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XXIV. On the structure of the atom: an investigation of the stability andperiods of oscillation of a number of corpuscles arranged at equal intervalsaround the circumference of a circle; with application of the results to thetheory of atomic structureJ. J. Thomson

To cite this Article Thomson, J. J.(1904) 'XXIV. On the structure of the atom: an investigation of the stability and periodsof oscillation of a number of corpuscles arranged at equal intervals around the circumference of a circle; withapplication of the results to the theory of atomic structure', Philosophical Magazine Series 6, 7: 39, 237 265

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T I l EL O N D O N , E D I N B U R G H , ANn D U B L I N

P H I L O S O P H I C A LAND

J O U R N A L O F

M A G A Z I N E

S C I E N C E .

[SIXTtt SERIES.]M A R C H 1904.

X X I V . On the S t ru c tu re o f the A to m : an [ .nvest lga tlon o f t lt eS t a b i l i t y a n d -P er i o d s o f O s c il la ti on , f a n u m b er o f C o ~Tus d esarranged a t equa l in terva l s around the Circumference o / 'aCi r c l e ; w i t h A p p l i ca t i o n o f tl~e r e s u lt s t o t h e Th eo r ~ o f A t o m i cS t r u c t u r e . By J. J . THOMSOn,F . R . S . , C a v e nd is h P r o f e ss o ro f E x p e r i m en t a l _ P h ys ic s, C a m b r i d g e 4T H E v i ew t ha t t he a t oms o f t he e l emen t s cons i s t o f anu m be r of ne ga t iv e ly e lec t r if i ed corpusc les enc losed ina sphere of un i form pos i t ive , e lec t r if i ca tion , sugges t s , am ongothe r in te res t ing math em at ica l p rob lems , the one d i scussed i llt h i s pape r , t ha t o f t he m o t ion o f a r i n g o f n nega t i ve l ye lec t r i f i ed par t i c les p laced ins ide a un i formly e lec t r i f i edsphere . Suppose wh en in equi l ib r ium the n corpusc les a rea r r anged a t equa l angu l a r i n t e r va l s r ound t he c i r cumf e r enceo f a c ir c le o f r ad i u s a , e ach co r pusc l e ca r r y i ng a c ha r ge e o fnega t i ve e l ec t r ic i t y . L e t t h ec h a r g e o f pos it ive e l ec t r i c it ycon t a i ned w i t h i n t he s phe r e be re, then i f b i s the r ad ius ofth i s sphere , the r ad ia l a t t r ac t ion on a corpusc le due to theposi t ive elect r i f icat ion is equa l to ve~a/ba; i f the corpusc les a rea t r es t th i s a t t r ac t ion mus t be ba lanced by the repuls ionexe r t ed by t he o t he r co r pusc l es . N ow t he repu l s ion a l ongO A , 0 be i ng t he cen t r e o f t he s phe r e , exe r t ed on a co r pus c le

e 2a t A b y on e a t B , is e q u al t o ~ c o s O A B ~ a nd ~ if' O A = O B ,e:t h is is equal t o 4O A 2 s i n~ A O B : hence , i f w e have n co r -

pusc les a r ran ged u t equa l ang ular in te rva l s 2 ~r / n r ound t hec i rcu m feren ce of a c i r c le , the r ad ia l r epuls ion on one corpusc leCommunicatedby the Author.P l d l . M - a t. S. 6 . gel . 7 . No. 39. M a r c h 1904. S

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2 3 8 P r o f . J . J . T h o m s o n on thed u e to t h e o t h e r ( n - 1 ) is e q u a l ( o

e ~ l 7r 27r 3~ r c os e c ( n - - l ) ~ r \~ a ~ c o s e c n + c o s e c - - + c o s e c - - + . . . +n nI f t h e c o r p u s c l e s a r e a t r e s t t h i s m u s t b e e q u a l t o t h e r a d i a la t t r a c ti o n . H e n c e : i f

2~r (n - - 1 )~"S , , = c o se c ~ + c o s e c - - + . . . c o s e cn n ~

v( :2a e2b - - ~ - . , ~ a a 2 S , ~ ,

or a 8 Snb~ - 4 v . . . . . . . . ( 1 )

T h e f o l l o w i n g a r e th e v a l u e s o f S~ f r o m n = 2 t o n - -- 6 .S ~ = 1 , S ~ = 2 " 3 0 9 4 , S ~ = 3 " 8 2 8 4 , S ~ = 5 " 5 0 5 6 , S c = 7 " 3 0 9 4 .

I n t h e i m p o r t a n t c a s e w h e n v -----n , i . e . w h e n t h e p o s i t i v ec h a r g e o n t h e s p h e r e is e q u a l t o t h e s u m o f al l t h e n e g a t i v ec h a r g e s i n t h e r i n g o f c o r p u s c l e s , w e g e t b y ( 1 ) th e f o l lo w i n gv a l u e s f o r a/b :~a

7 2 . ~ .

2 . . . . . . . . . " 53 . . . . . . . . . " 5 7 7 34 . . . . . . . . . " 6 2 0 85 . . . . . . . . . " 6 5 0 56 . . . . . . . . . " 6 7 2 6I f t h e r i n g o f c o r p u s c l e s , in s t e a d o f b e i n g a t r e st , i s r o t a t i n gw i t h a n a n g u l a r v e l o c i t y ~ , t h e c o n d i t i o n f o r s te a d y m o t i o n i s

y e 2 a e 2b -w = mato~ + ~ S~,,qk~ nr va ~ m ~ + _ ;

- ~ - = e ~ 4h e r e m i s t h e m a s s o f a c o r p u s c l e .W e s h a l l n o w p r o c e e d t o f in d t h e f o rc e s a c t in g o n ac o r p u s c l e w h e n t h e c o r p u sc l e s a r e s l ig h t l y d i s p la c e d f r o m~ h e ir p o s i t i o n s o f e q u i l i b r i u m . L e t t h e p o s i t i o n o f t h ec o r p u s c l e s b e f ix e d b y t h e p o l a r c o o r d i n a t e s ~ - a n d 6 i n t h ep l a n e o f t h e u n d i s t u r b e d o r b i t, a n d b y t h e d i s p l a c e m e n t z a tr i g h t a n g l e s t o t h i s p l a n e ; l e t r~ , ~9,, z , b e t h e c o o r d i n a t e s o ft h e s t h c o r p u s c l e ; t h e n , s i n c e t h e c o r p u s c l e s a r e b u t s l i g h t l yd i s p l a c e d f r o m t h e i r p o s i t i o n s o f e q u i l i b r i u m , r , = a + p ,w h e r e p , i s s m a l l c o m p a r e d w i t h a, z, i s a ls o s m a l l c o m p a r e dw i t h a , a n d t ? ,- -t g , _ x ~ 2 ~r + ~ b , - - r w h e r e n i s t h e n u m b e ro f c o r p u s c l e s a n d t h e ~ b's a r e s m a l l q u a n t i t i e s .

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S t r u c t u r e o f th e A t o m . 239Th e radial repulsion e xerted by the sth corpuscle on th eTth is equal to

d 1- - e 2 a r p ( , ; + ~ , : - ~ , . , . c o s~ O ~ - o , , ) + % - , y ? ;

exp and ing this, reta inin g only the first powers of p, ~b, and z,we find that if Rp, is this repulsione~ p. ( 3 i

_ ~ l '1 + 1 ) --~(r cot + }2 s i n 2~he,-e ~ = (p-~) ~.

The tang ential force Op, tend ing to increase 0p is equal toe~ 1

~ / o , t " ; + ~ ' ; - ~ r , , , c o s ( % - 0 ~ ) + ( ~ - - ~ . ) ' ~ } ~ ;expa nding this and retain ing o nly the f irst powers of theslnall quantities, we get

e ' c o s + f 3 0 , 1 p , ( q b , - - ~ , ) ( e o t ~ + ~ t a n ~ ) }Opt= 4a2sin ~ ~ ( 1 2 a 2 aZp,, the force at r igh t angles to the undisturbed plane ofthe orbit , is easily seen to be give n by the equation

e 2

T he total radial force I/p exerted on the p th corpuscle byall th e o the r corpuscles, is equal toe~ S - - A / - -E A , a~ B~ Po pp+. ~ . p . - d p p + , p .p + ,,

where S = ~ -k 1 1s i n ~ r s i n ~ + . . .. in ( n - - 1 ) ~ ' ;

n n n

A' e~ [ 3 [ i I i \- - s m - s i n ( n - -n n 7~

1/" I 1 1sln -- Sln ~ s in 3

n // /~$ 2

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"240

, /gC O S ~ .

C 8

P r o f . J . J . T h o m s o n on t~ee"~{ l 1 ]

A ~ . p + , = o a = \[z '~-a tsi-~ . + - - - ] ;n8 ' / r

e 3 c o s - - nB v. v+ ~ = ~ a ~ s in~ s ~ '"n

T i l e c oe f f i c i e n t o f ~ bp i n th e e x p , r e s s i o n f o r R p v a n i s h e s ,s i B c o S q rc o sV n = 0 .

si n .~ 5? lA s A p .p + ,, B p .r +~ d o n o t i n v o l v e p , i t is m o r e c o n v e n i e n tt o u s e t h e s y m b o l s AS a n d B . f o r t h e s e q u a n t i ti e s , a n d t ow r i t o ] R p = ~ ' a S - - p ~ A I - - Z P p + ~ A ~ - a~ bp +~ B~ .T h e t a n g e n t i a l f o r c e O p a c t i n g o n t h e p ~ th p a r t M e m a ys i m i l a r l y b e w r i t t e n

O ~ = X pp+ ,B~ - - a ~ pC + a X ~ bp+ ,C .w h e r o27r

n$ , wc o s - -e n / .s ~r 1 . s ~ \.--,~ ~ / co t - - + ~ ta n - | ;

~ a ' s i n , s ~ \ ~ z n /nw h i l e Z p, t h e f o r c e a t r i g h t a n g l e s t o t h e p l a n e o f t h e o r b i t , i sg i v e n b y t h e e q u a t i o n

Z p = z pD - - X z p + ~ D .w h e r e D = e: { 1 1 1 ")~ + . - - ~ ,4 ( , ~ - 1 ) ~r ] '8 a a \ s i n a_ ~ s in S _ _ ~ s i n ~7t gt ~ ' ta n d

n

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S h ' w c t a r e o f t h e A t o m . 241Th e equations of m ot ion of the p t h corpuscle a re

" ' ~ , @ - " ~ , a t J ! = - - - - U I ~ ; . . ( ~ , )/ d , o , ~ a~ a o , ~ = % . , . ( ~ )(g~,7 V ~2

R e t a i n i n g t u f t y t h e f i r s t p o w e r s o f s m a l l q u a n t i t i e s ~ w e g e td 6f rom these equat ions , if~co s the value of ~ w hen the m ot ionis stead)-,y e ~ a o= m a e o ~ + 4 " a2 S ~La

d ~ _ 2 m a w d fl ~ = p , ( m o J ~ _ v e~ '~ e 'm dt .~~ b~ ] + R e - - l a y S .IE pp an d 8p va ry as e~e, this equat ion m ay be w ri t ten

( & - - m ~ l ~ ) p p + AIpp+I+ A:pp+:+ . . .- - 2 m a o J , q d ? , + aBa(~e+l+ a B :r + . . . . O ,where

e: { ( 1 1 ) ( ~ l ) }A = ~ 8 + A ! = e ~ 5 + ~ + . . . . + ...s i n - +in - -n n n n

W r i t i n g 1 , "2 , 3 for iv we g e t( A - m , f )p , + A lp~ + A~pa ... + A , ~ - i p , , - - 2 , n a o J , 7 * , + aBd~s + aB ,q ,a + . . . . 0 1( a - , , , v ' ) p = + & p ~ + a , m + " . . . . . . . . . - 2 , , , ~ , , q e p , . + ~ s , + ~ + ~ s , e e ~ + . . = o [ ( a )

)A - - m q ~ ) p ,~ + A , , o , + A ,P .2 + . . . . . . . . . - - 2 m a o a u t ~ , , + a B ~ + a B . ~ t . . . = 013y equ at ion/3 we h ave

2,n~ov/~ - - B~PP+Aa - - B .,. P'+~a + ' " .(C - m q s) 4 ' v - C~q~v+~--C ~ d? e+ 2 - ' " = O .W ri t in g 1 , 2 , 3 in success ion for p we ge t

p2a - -B , paa - - B . ~ . . . + (C--mq ~ = 0 { (B),~mwq

2~ , , ,~ ,~ P ~ - B ~ a t _B . ,m .7 . . . . .( C - , , , , f ) q , , , - ( ~ d , : - c d , ~ - . = o

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2 4 2 P r o f . J . J . T h o m s o n on theT o s o l v e e q u a ti o n s A a n d B w e n o t i c e t h a t i f w b e a n yr o o t o f t h e e q u a t i o n x " = l , i . e . i f o~ b e o n e o f t h e n t h r o o t so f u n i t y , e q u a t i o n s A w i l l b e sa t is f i e d b y

p ~ = ~ p l , p ~ = ~ p , , p , = o , p , . . . r r 1 6 2 ~ , = ~ . .p r o v i d e d

Pl (A -- mq ~ ~A I + eo=A= .. e ~ n - A n - l )+ ~ b ,a ( - - 2 , m e o r + ~ B 1 + to 2B 2 + t o " - ' B ~ _ , ) = 0 ; ( 1 )

w h i l e e q u a t i o n s B w i l l b e s a t is f ie d b y t h e s a m e v a l u e sp r o v i d e dpl(2em~q - ~ B 1 - - t o 2 B z - r 1 )+ ~ ( c - , . r = o . . ( 2)H e n c e , i f b o t h s e t s o f e q u a t i o n s a r e s a ti sf ie d b y t h e s e v a l u e s ,w e h a v e , e l im i n a t i n g p l a n d 4 x f r o m ( 1) a n d ( 2) ,

( ( A - m q ~) + ~ , A , + o ,'A = + . . o,~ - 1 . ~ , , _ 1 )( C - m ~ , ~ - ~ ,C 1 a , 2 C ~ - ~ o " - lC , , _ l )= - - ( - - 2 t m ~ q + O ) B l + o ~ 2 B ~ + . . eon-X B,)~ , 9 (1)

a b i q u a d r a t ic e q u a t i o n t o d e t e r m i n e q t h e f r e q u e n c y o f t h eo s c i l l a t i o n s o f t h e s y s t e m . : N o w o~ i s o f t h e f o r m2kTr 2klrc o s - - + t s in - -n n "*

w h e r e k i s a n i n t eg e r b e t w e e n 0 a n d n - 1 . S u b s t i t u t i n gt h i s v a l u e f o r t o , w e f i n d e2 fcos2k~r ( 1 + ~ )s in -

. . . . . n //

s i n - - _ _ . . . . . n s i n - - - -n n n T t

W e s h a ll d e n o t e t h is b y L k ; i t w i l l b e n o ti c e d t h a t L k c o n -t a in s n o i m a g i n a r y t e r m s . W e f in d a ls o t h a t' T r

~ C l + ~ C ~ + o ~ C ~ + e o ~ - ~ C " - a = c o s - - - c o t ~ + t a n.n.. sln~ Kn

27r+ c o s 4k~r" c~ "-n-"( c o t ~ -~ + ~ . a n ~ )

s i u - - - ; ~ " 'n

\

W e s h a l l d e n o t e t h i s b y N k . + "" " ) "

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Structure of the Atom. 243A g a i n , 27rt . e~ [ . 2k~" 4kTr cos --no~B1 + oo2~B2+ to~'-lBn_l----- ~ ~ s l n - - + si n 2~rn n sin~ __

7FCOS --nsin2 Trn

3~"+ sin 6 k~r cos n )

n sin , 3Trn=iM k , s a y .

Subst i tu t ing these values , equat ion (1) becomes( ( A - - m q 2) + L k ) ( C - m q ~ ( M k - - 2 m a , q) ~. . ( 2 )

From the value of C given on p . 240 we see tha t C is thevalue of Nk wh en k----0~ an d so m ay be denoted by No, and3 e~ S - L 0 ; hence equa t ion (2) m ay be wr i t t enhat A = ~ a~

3 e2 - m 9 ~) (No--N k--mq ~) = (M~--2mo, q) ~. (3)k in th i s equa t ion ma y have any va lue f rom 0 to (n - - l ) ; bu t wesee th at i f w e w rite n~--k for k, t i le values of q given by the tw oequat ions d if fer only in s ign , and so g ive the same frequencies ;n - - 1thus a l l the va~tues o f q can be go t by pu t t i ng k = 0 , 1 , . . . 2ni f n b e o dd , o r k = 0 , 1 i ~ i f n h e e v e n ; t h u s i f n b e od d th e ren + la re ~ equa tions o f the type (3 ). W he n k = 0 , M~=0~ and(3) reduces to a quadra t ic equa t ion ; so tha t the nmn ber o froots of these n + 1 n + 1- ~ - e q u a t i o n s i s 4 x ~ - - 2 = 2 n ; i f n be

neven the re ar e ~ + 1 equ atio ns; bu~ as M~-----0 wh en k----0n two of these reduce to qua dra t ics ; so tha t thea n d k = ~ , kl lunxber of roots o[ these equa t io ns i s ~(~ ,, + 1)- -4- ~-~ n.

Thus in each case the num ber of roots is equal to 2n , thenmnber of degrees of f reedom of the corpuscles in the p laneof the i r und is tu rbed o rb it .Le t us now consider the motion a t r igh t angles to th is p lane .By e q u a t io n q we h a v e _d~z~. ve~m -d i ' - = - - -~- :p + D zv - ~D.zv+~ ;

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2~4 Pr of '. J . J . Th om son o n t h eo r i f : v i~ prop or t io na l to e qt,

- { ve~ - D - - m q + ) + Z D , z p + ,= 0 ;t h u s - , , , i ' ) = o , Izl \b ~ - - D + z ~ D l + z 3 D 2 + . . . [

.~Z~--D-r e - m q ~ . . . . O, ~' . . ( C )I

z , , ~ b ~ - - ' - - " 7 ) + z l D , + z 2 D : + . . . = 0 . _ jW e ~ e t h a t to a g a i n b e i n g o n e o f t h e n t h r o o ts o f u u i t y , t h es o l u t io n o f e q u a t i o n s C i s

Z 2 -~ O.)Z1 ~ Z 3 "-- toZ 2, Z 4 ~ 0).7 ,;~.. .Ye 2 ~ ob3- - l ) - m f f ~ + o l ) ~ + o ~ D ~ + t o ' - ~ D ~ - l = 0 . . . ( 4 )

2k~r 2k~rP u t t in g t o = c o s - - + , s i n - - - , w e f in d, s u bs ti t u t in g th ev a l u e s f o r D g i v e n a b o v e , t h a t 2k~" 4k~r 6k~re2 / c o s - c os c o s - -n {_ _ _ _ _ f _ n n

t o D l + ~ ~ 1 7 6 ~ t s i n ~ ~ s in 3 2 , r + s in ,~ ~ r9~ n n

)D e n o t i n g t h is b y P ~ a n d n o t ic i n g t h a t D = P o , w e f in d t h a te q u a t i o n ( 4 ) b e c o m e s

~ 6~,~ + P ~ - - P 0 - m q ~= O .

P u t t i n g i n s u cc es sio n k = 0 , 1 , . . . n - - I , w e g e t n v a lu e so f q g i v i n g t h e n f r e q u e n c i e s c o r r e s p o n d i n g t o t h e d i sp l a ce -m e n t s a t r i g h t a n g l e s t o t h e p l a n e o f' t h e u n d i s t u r b e d o r b i t .W e s h a ll n o w p r o c e e d t o c a l c u l a te th e f r e q u e n c i e s f o rs y s t e m s c o n t a i n i n g v a r i o u s u u m b e r s o [ c o rp .u s cle s. T h e f o u rq u a n t i ti e s L ~ , M k , : N k , P k w h i c h o c c u r m t h e f r e q u e n c ye q u a t i o n m a y b e e x p r e s s e d i n t e r m s o f t h r e e q u a n t i ti e s S ~-,Tk~ U k, w he r e2 k s ~ r 1~ =~ = ~ 1 C O S n s in s ~ '

n

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Structure of the Atom.2 ks~r 1

k ~-~- .5 CO S

1 n sins s~r '~x~sTr cos - -U ~ = Z ~ s in n ,"

sin~ s_~fo r we have

e 2Lk = (S~ + Tk) ~ ,o]l{k= U k ~a3,

Case of two corpuscles.W h e n n : 2 w e h a ve2e~L o = 8 ~ , M o = 0 ,

2e~L1 = - ba-=S, M 1 - 0,

Nk = (2T;. -- Sk) ~ a ~,e 2P ~ = T ~ ~ a ~.

245

N o = 8 ~ , P o = ~ /~ ,o e2e ~N ,= ~5@' PI = - ~ "

He nce fo r v ib ra t ions in the p l ane o f t he o rb i t we have ,wh en k = O,(3 e:_~s -mq2)(--mq")-=4m:~~ " ;

the roots of th is equat ion areJ . / 3 . e ~3 e~ +4~o~=4r ~Tb3 +co".q= 0, ~ = ~: ma

When k - - -1 , t he f requency equa t ion i s4 a "~ --mq-) =~m'~o-~t ;

the roots of th is equat ion are/ 1 e~ .- ~ / ~ --m~ +" =~ '- +- ' V ,7~t;

a n d J i e~ / ,,%~"_ - - +~ =- a , + ~ / ,~b3,= - - a ~ + - 4 ma ~ __

the ~econd se t of va lues only d i i tbr ing in s ign from the f i rs t .

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2 46 P r o f . J . J . T h o m s o n on theF o r t h e v ib r a ti o n s p e r p en d i cu l a r t o t h e p l an e o f t h e o r b it ,w e h a ve f or k = 0 , / ~ - ~ .

q = % / . -T g . 'fo r k = l ,V m b a 4 m an : o J .

Thu s the six f r equenc ies c or respo nding to the s ix deg reesof t>eedom of the two corpusc les a re%//~-7. / , e r / ~-~ / : ~0, f O ~ ~ , % /gb~ -o , , % /~ ,~ + o,, % / ~ o,,.

W h e n t h e c o rp u sc le s a r e n o t r o t a t in g r o u n d t h e ci rc l e ,/ vet wo oF t h e s e r o o ts a r e z e r o , t h r ee eq u a l to ~ / ~ , an d . t h e

s ixth equ al to "V ~b~, " Th us the ef fect of ro ta t ion on thet r i p l e f r eq u en cy v / ~ m b ~ i s to sepa rate the roots , one re-m a i n i n g u n a l t e r ed , o n e i n c rea s i n g , an d t h e o t h e r d i m i a i s h i n g .

Case of three corpuscles.W h e n n = 3 .4 1 6 2 8 e-"S o - , / ~ , T o= 3 U ~ ' U o = o , L o= g / : ~ 8an,v v20 e= 16 e~

$ 1 = 2 T I = - - 8 ~ -~3 1.4 e :- 7~:~' a v / ~ , u l = _ _ , , L l = ~ v/~ S a~ ,1 0 e 2 2 e 2 8 e ~~ 1 = - a v / ~ s~~' M I = 7 / ~ ~ a~ ' t ' i = a v / ~ Sa"

$ 2 = $ 1 , T~ = T , , U ~ = - - U 1 , L ., = L1, : N ~ = N ,M ~ = - -M 1 , P ~ = P i .F o r t h e v i b ra t io n s i n t h e p l an e o f t h e o r b it , wh en k = 0 ,t h e f r eq u en cy eq u a t i o n i s

the solut ion of th is i s{ " ~ ~ (3ve~~ } 89

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Structure o f the A tom. 27W h e n k = 1 , t he f r equ enc y equa ti on is

The so lu tion of th i s e qua t ion i s~ / / ~ e~ Iv-'eq-,q = ~ + ~ m ~ + ~ + % / " m-5~

q = - -r + t o ~ = - -~ o +~a .~ - ~ m b ~

W h e n k = 2 t h e f r e q ue n c ie s a r e t h e s am e a s w h e n k = l ; w ehave thus s ix f r equencies cor responding to the s ix degreeso f f r eedom o f t he t h r ee co r pus c l e s i n t he p l ane o f t he i rund i s t u r bed o r b it .Fo r t he v i b r k ti on a t r i gh t ang l e s t o t he p l ane o f th i so r bi t, w h e n k = 0 t h e f r e q u e n c y e q u a ti o n is

l,e 2b- -- m 7~= 0,o r / v e~q = % / mb-~"

W he n ] r 1 , t he f r equ enc y equa t ion isi,,e ~ e 2b~ x / g a 3 m q ~ = 0 ,o r q =

In th e case o f th ree corpusc les, as in tha t o f two~ we seet ha t w he n t he r e is no r o t a ti on t h r ee o f t he pe r i ods a r e equa l ;t he s e a r e s epa r a ted w hen t he co r pus c le s a r e i n r o ta t ion .Case o/ fo ur corpuscles.W he n n-- - 4 ,

e 2S o = l + 2 v / 2 , T 0 = 4 V / 2 + l , C o - O , L o = ( 6 v / 2 - + 2 ) S a - ~ ,e~ r

e 2S l = - - 1, TI= - -1 , U I : 2 v / 2 , L l = - - 2 ~ a3,e ~ 6 2 e ~

NI= 8a3, M I = P V Q ~ a L P 1 = - 8 - 3 ,e 2

S ~ = - - 2 v / 2 - + 1 , % = - 4 v / i f + l , U ~ = 0 . L p = ( - -6 v / 2 - + 2 ) ~ a ~ ,e ~ e 2

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248 Prof . J . J . Thomson on theW he n k = O, thB frequ ency equation is

(1+ ~J~) -m q ~) ( - . , r ~,,e..,,q,;the solution of which is

q = ~ / 3 e~ / i~ve~q = 0 ,W h e n k = l , t h e f r eq u e n c y e q ua ti on is

the solution of this isq = + / - v 71q = - r o + + 1 e~ + r ~ = / I v / i f + 1 ve~

~7,--~ - - z + N / ~ 1 mb~When k=2 , the f requency equa t ion i s

4m~w~9 e.

l~egarding this as a quadratic in qe, we see that the roots areposit ive, so that the values of q are real and the arrangementis stable. The roots of th e equation are3 (r162 ~~ = ~ / 2 ~aa~ + 2 ~

e ~v 128 v~ a ~ + 2 r ?"a~

Let us now consider the mot ion a t r ight angles to theplane of the orbi t . W hen k=0 ~ the f l 'equency equat ion isub; V -- ~n~ ~ = O,

%/~ib'"W he n k = 1, the frequ ency equation is

9 e 'Z e ~z ,' - (~ v ' ~ + ~) ~ _, , ,~ . - , = o ,

or , 1= + t o .

2 v/2 + 1

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o r

T h u s , u n l e s s

S t m w t u r e o f t h e A t o m .W h e n k = 2 , t h e f r e q u e n c y e q u a t i o n i s

r e ~ 8 ~ / 2 e ~U - - ~ - - m q . ,= O ,8 r ~ ( 4 r v e ~' ? - 4 , / ~ + ~ , - ~ -r + s ,,,~,--~:"

2 4 9

q2 i s n e g a t i v e , a n d t h e e q u i l i b r u m i s u n s t a b l e , t h e f o u r c o t .p u s c l es t h e n a r r a n g i n g t h e m s e l v es a t t h e c o r n e r o f a r e g u l a rt e tr a h e d r o n . W h e n , h o w e v e r , ~ is l a r g e e n o u g h t o s a ti sf yc o n d i t io n ( 1 ) , f o u r c o r p u s c l e s w i l l b e i n e q u i l ib r i u l n w h e ni n s t e a d y n m t i o n i n o n e p l a n e a t t h e c o r n e r s o f a s q u a r e .

C a s e o f f i v e c o r p u s c le s .W h e n n = 5 , w e h av eS o - = 5 " 5 0 5 6 , T o = 1 2 " 1 7 3 2 , U o = 0 ,

eN o = ~ ( 1 8 ' 8 t 0 8 ) , M o = 0 ,

S t = - - - ' 6 5 , T t = 1 " 1 6 0 9 , U 1 - - ' 4 " 8 5 6 ,

2L o = ~ ( 1 7 . G T S 8 ) ,P o = - [ : ~ ( 1 2 . 1 7 3 ~ ) ,L,= f~ ( .su) ,

~ ge 2S ~ = - 2 "1 0 3, T , = - - 7 " 2 4 9 , U ~ = 2 . 1 0 3 , L 2 --= - ~ ( 9 .3 5 2 ) ,

~? e ~ e uN ~ = - -~ w ~ 1 2 '4 , M 2 = ~ 2 " 1 0 3 , P ~ = -- ~d ~ 7 "2 1 9 .T h e f i- e q u e n c y e q u a t i o n w h e n k = 0 i s

5"5056 ~ - - m q K - - m q ) = i m ~ a ~ 7 ~e l , ' - ] -

the s o l u ti o n o f w h i c h i s / 3 r e - ~q = o , - q = % / ~ - + ~, .W h e n k = l , t he f r eq u e n c y e q u at io n is

( 15 .8 7 e~ _ m ~ f / e : ,~, : , = ~ , S - = , 4 . 8 5 6 - 2 , , o , q ) ,

. . , (1)

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2 5 0 P r o f . J . J . T h o m s o n o n t h e# I . ' ,o r q = o ~ + 1 0 ' 9 1 8 ~ -i- ~o'-= ~ + m b ~,

W h e n k = 2 ,e~ 2 P = ( 2 . 1 0 3 8 a ~ _ 2 m t o q ) .

B y a p p l y i n g t h e u su a l m e t h o d s w e f in d t h a t a ll t h e ro o t s o ft h i s e q u a t i o n a r e r ea l , s o t h a t t h e s t e a d y m o t i o n oE t h e f i v ep a r t i c l e s is s t a b l e f o r d i s p l a c e m e n t s i n th e p l a n e o f t h eo r b i t .L e t u s n o w c o n s i d er d i s p l a c e m e n t s a t r i g h t a n g le s t o t h ep l a n e o f t h e o r b it . W h e n k = O t h e f r e q u e n c y e q u a t io n i s

v e 2b-=~ - - m q ' ~ = O ,

t h e s o l u t i o n o t~ w h i c h i s

W h e n k = i , t h e f r e q u e n c y e q u a t io n is~ q 't ~ ~ - m ~ 2 ~ 0 7h e n c e

W h e n k = 2 , t h e f r e q u e n c y e q u a ti o n i s~ - -- 1 9 "4 2 ~ - - m q ~ = O ,

o r 19"42ma~2 + v e ~ 1 9 ' 4 2 v e ~1 --1 - - ~ - - 1 1 b3 m q ' = O ,

1 9 " 4 2 ~ 8 " 4 2 v e ~- 1 1 - m r ~ 1 1 b 3 - - m q : = O .

H e n c e , i n o r d e r t h a t t h e e q u i l i b r i u m m a y b e s t a b l e ,~ m a s t b e 8 "4 2 v e 2 v e ~> " 1 9 ,4 ~ b > ' 4 3 3 ~ .

T h u s t h e f iv e c o r p u s c le s a r e u n s t a b l e w h e n i n o n e p l a n eu n l e s s t h e a n g u l a r v e l o c i t y e x c e e d s a c e r t a in v a l u e ; t h ea r r a n g e m e n t i s s t ab l e , h o w e v e r , w h e n t h e a n g u l a r v e l o c i t yi s l a r g e .

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Structure of the Atom . 25 1Ca.~e o f s ix corpttscles.W h e n n = 6 ,

S " 4 1 6 ( 2 2 2 8 \ e "o = a + - ~ , T o = 1 7 + ~ v / 5 , U o = 0 , L o = + ' - ~ ] s a . ,a~g)g-~ N 0 = o , P o = +

s~ = - , / I 3 ' T 1 = 7 - a , / g , u 1 = G + ~ , T ,, = 8 - 5 ~ / ~ 8 , ~ ,/ 10 \ e2 8 e: / 8 e~

2 8 2 = / - 8 14 ~1 e"S , = - -1 - - x /~ '- - T~ = - - 7 - - --3v/~ , U. .- -- 6 - - ~ , L= \ - :3 ~ ] ~ a i i ,/ 10 " ,e ~ [ ~ 2 , , e ~ P 2 = [ - 7 8

4 16S a ' - - - - 5 + ~ - : ~ , T a - - - - 1 7 + ~ , U a---0 , L a - - ( - 2 2 . . I - 28 "~e~~ T ~ = ( - - 2 9 + ~ ( \ ~ e = = ( - - * r + 1 6 ~ e :

I t is n o t n e c e s s a r y t o w r i t e d o w n a l l t h e f r e q u e n e y e q u a .t i o n s b e c a u s e , a s w e s h a l l s h o w , t h e a r r a n g e m e n t o f s i xc o r p u sc l es i s u n s ta b le '. F o r w h e n k = 3 t h e f r e q u e n c y e q u a -t i o n i s( 6 ( 5 + _ f , e= e ~ o e ~4/~ , . , --4 4 8 . -m t]" )(5 8 ~ --'mq2)=lrn%o'~t"-,

A s 1 1 - - 8 ~ / 3 i s p o s i t iv e , w e s e e t h a t o n e o f t h e r o o t s o f t h i seq ua t i o n fo r q~" i s ne ga t i ve , so t h a t g i s im ag in ar y ; . t h i s showst h a t t h e s t e a d y m o t i o n o f 6 c o r p u s c l e s i n a r i n g ~s u n s t a b l e ,h o w e v e r r a p i d t h e r o t a t io n . W e c a n , h o w e v e r , m a k e t h em o t i o n s ta b l e b y p u t t i n g a c o r p u s c le a t t h e c e n t r e ; i f w eh a v e a n e g a t i v e c h a r g e e q u a l t o t h a t o f p c o r p u sc l e s a t t h ec e n t r e o f t h e r i n g t h e r a d i a l f o r c e i t e x e r t s o n t h e s t h9 P ~ Pe ~ 2 p e ~ pc o r p u s c l e l s ( a )2 ~ o r ~ - - a~ I n t r o d u c i n g t h is t e r mi n t o t h e e x p r e s s i o n f o r t h e r a d i a l f o r c e w e f i n d t h e f r e q u e n c y

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2 5 2 P r o f . J . J . T h o m s o n o n t h ee q u a t i o n " b e c o m e s

3 e ~ 3 p e ~" 2 :U s i n g t h is f r e q u e n c y e q u a t io n , a n d s u p p o s i n g t h a t p - - l ,i . e . t h a t t h e r e i s o n l y o n e c o r p u s c l e a t t h e c e n t r e o f t h eh e x a g o n , w e g e t i n s te a d o f ( 1 ) ,

T h e r o o t s o f t h i s e q u a t i o n i n 9 ~ a r e b o t h p o s i t i v e , so t h a ti s r e a l a n d t h e e q n i l i b r i m n i s s t a b l e .L e t u s n o w i n v e s t i g a t e th e c o n d i t io n s f o r s t a b i l i ty f o rd i s p l a c e m e n t s a t r i g h t a n g l e s t o th e p l a n e o f t h e o r b i t .F o r t h e m o t i o n a t r i g h t a n g l e s t o t h e p l a n e o f t h e r i n g ,t h e f r e q u e n c y e q u a t i o n w h e n k = 3 is

v e ~ p e ~ 3 4 e ~U - - -

F o r t h i s t o r e p r e s e n t t h e d i s p l a c e m e n t o f a s t a b le s y s t e m 9 ~m u s t b e p o s it iv e , s o t h a t i f p = 1v~ .~ e ~ 3 4 e ~b - - - a - - ~ - $ a ~

m u s t b e p o s i t iv e ; w e h a v e , h o w e v e r ,r e : e ~ e ~ 4 )+

s o t h a t f o rv e ~ e ~ 3 4 : e ~I ,T - - ~ - - ~ ,c ?

t , ) b e p o s i t i v e1 " 2 - 4 : / ~ 3 v e ~ . v e ~m oo~ m u s t b e g r e a t e r t h a n ~ ~ , ~. e . " 4 6 / 7 .

L e t u s n o w c o n s i d e r t h e s t a b i l i ty o f t h e c o r p u s c l e a t t h ec e n t r e o f t h e r i n g : i f i t i s d i s p l a c e d t h r o u g h a d i s t a n c e z a tr i g h t a n g l e s t o t h e r in g ~ t h e e q u a t i o n o f m o t i o n o f" t h ec o r p u s c l e s i sd~ z ve ~ 6e '~

T h u s i f t h e m o t i o n i s s t a b l ey8 2 ~e $b" > a:'

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S t r u c t u r e o f t]te A t o m . 2 531 5 - - 4 / ~ / - 3 r e " . . v e ~o r m~ 2 t b:~ ,i. e. "o 3-- ~- .

This value of o~2 i s g re a te r than tha t r equ i red to. m ake thee q u i l i b r im n o f t h e r i n g s t a b le f o r d i s p l a c e m e n t s a t r i g h tangles to i t s p lane ; i f the cenf i ra l corpuscle , ins tead os beingi n t h e p l a n e o f t h e r i n g , w a s o n e s id e o f t h e c e n t r e o f t h esphe re o f pos i t ive e lec t r i fi ca t ion wh i le the r ing was on theo t h e r si de , t h e r o t a t io n r e q u i re d t o m a k e t h e e q u il i b r iu m o ft h e d e t a c h e d c o r p u s c l e s t a b l e w o u l d b e l e s s t h a n w h e n i t w a si n t h e p la n e o f t h e r i n g ; f o r e q u i l i b r iu m t h e d is t a n c e o f t h ed e t a c h e d c o r p u s c l e f r o m t h e c e n t r e o f t h e s p h e r e m u s t b e s i x~ imes the d i s t ance o f the p lane o f the r ing f rom tha t po in t .

C o n d i t i o n s f o r t h e s ta b i li ty o f r i n g s c o n t a i n in g m o r e t h a ns i x co~Tusc l e s .I f ind tha t a s ing le co rpusc le in the cen t r e i s su f f ic ien t tom ak e r ings o f 7 and 8 co rpusc les s t ab le ; in the l a t t e r case ,how ever , one o f the va lues o f q~ thoug h pos i t ive is excee d ing lys m a ll . W h e n th e n u m b e r o f c o r p u s c le s e x c e e d s 8 t h e n u m b e ro f ce n t r a l co rpusc les r equ i re d to ensu re s t ab i l i ty inc reasesv e r y r a p i d l y w i t h t h e n u m b e r o f c o r p u s cl e s i n t h e r i n g .T h e f r e q u e n c y e q u a t i o n i s

3 P e~ _ ~" ~ ~ __

N o w N o - N k i s a l w a y s p o si ti v e a n d M is s m a ll c o m p a r e d w i thL and N; hence th i s equa t ion w i l l have r ea l roo t s i f3 e~S0 3 p e ~4 a ~ + ~ ; - - (L ~

i s p o s it iv e . T h e g r e a t e s t w d u e o f L 0 - - L ~ i s g o t b y p u t t i n gk = n / 2 w h e n n i s e v e n , a n d = ( n - - 1 ) / 2 w h e n n i s o d d : h e n c ethe cond i t ion th a t the va lues o f q shou ld b e r ea l , i . e . tha t thee q u i l i b r i u m o f t h e r i n g s h o u l d b e st a b le , is3 p e ~ 3 e'~S~,a~ > ( L 0 - L . ~ ) - ~ - ~ - w h en n is e ve n,

a n d3 p e ~ ( L o - - L . - 1 ) 3 e ~S 0 w h e n n i s o d d .

F r o m t h i s e q u a t i o n w e c a n c a l c u la t e t h e l e a s t v a lu e o f pwh ich w i l l m ake a r ing o f n co rpusc les s t ab le . The va lues o fP h i l . M a y . S . 6 . Vol. 7 . No. 59 . M a r c / t 1904 . T

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2 5 4 P r o f . J . J . T h o m s o n on theP fo r a s e r ie s o f va lues o f n a r e g iven in the fo l low ingt a b l e : - -

n . . . 5 6 7 8 9 10 15 20 30 40p . . . 0 i 1 I 2 3 1 5 3 9 1 0 1 2 3 2F o r l a r g e v a l u e s o f n t h e v a l u e s o f p a r e p r o p o r t i o n a l ton 3. W h e n p i s g r e a t e r t h a n o n e , t h e i n t e r n a l c o r p u s c l e sn e c e s s a r y t o p r o d u c e e q u i l i b r i u m c a n n o t a l l b e a t t h e c e n t r eo f the sphere , the y w i l l s epara te un t i l the i r r epu l s ions a reb a l a n c e d b y t h e a t tr a c t io n o f t h e p o s i ti v e e l e c t r i c i ty i n t h es p h e r e . T h u s w h e n t h e r e a r e t w o in t e r n a l c o r p u s c le s , a sw h e n n = 9 , t h e s e t w o w i l l s e p a r a t e a n d w i l l f o r m a p a i r w i t h

t h e li n e j o i n i n g t h e m p a r a l le l t o th e p l a n e o f t h e r i n g . I f w ea s s um e , a s i s a p p r o x i m a t e l y t h e c a se , t h a t t h e p a i r o f e q u a lco rpusc les ex e r t s a t ex te rna l po in t s the s ame fo rce as a doub lec h a r g e p l a c e d a t a p o i n t m i d w a y b e t w e e n th e m , th e p r e c e d i n gtheory w i l l app ly~ and the sys tem cons i s t ing o f the r ing o f 9a n d t h e p a i r o f c o r p u s c le s w i l l b e i n s t a b le e q u i l i b r i u m .W h e n n = 1 0, t h e in t er n al c o r p u s cl e s m u s t b e t h r e e i n n u m b e r ;t h e s e t h r e e w i l l a r r a n g e t h e m s e l v e s a t t h e c o r n e r s o f a nequ i l a te r a l t r i ang le , and th e sys te m o f 13 co rpusc les w i l lcons i s t oF a r ing o f 10 and a t r i ang le o f 3 , the p lanes o ft h e r i n g a n d t r ia n g l e b e i n g p a r a l le l b u t n o t c o i n c i d e n t ; t h ec o r p u s c l e s a r e a l l s u p p o s e d t o b e i n r a p i d r o t a t io n r o u n d t h ed i a m e t e r o f t h e s p h e r e d r a w n a t r i g h t a n g l e s t o th e p l a n e s o fthe r ing . F o r a r ing o f 12 co rpu sc les we r equ i r e 7 in side~bu t 7 co rpusc les , a s we hav e s een , cann o t fo rm a s ing le ring ,b u t w i l l a r r a n g e t h e m s e l v e s a s a r i n g o f 6 w i t h o n e a t t h ecen t r e . Th us the sys tem o f 19 co rpusc les w i l l cons i s t o f anou te r r ing o f 12 , an inner r in g o f 6 in a p lane pa ra l l e l to theo u t e r r i n g , a n d o n e c o r p u sc l e a l o n g t h e a x i s o f r o t a ti o n .In . t h i s w a y w e s e e t h a t w h e n w e h a v e a l a r g e n u m b e r o fc o r p u s c le s i n r a p i d r o t a ti o n t h e y w i ll a r r a n g e t h e m s e l v e s a sf o l l o w s : - - T h e c o r p u s c l e s f o r m a s e r i e s o f r in g s , t h e c o r p u s c l e si n o n e r i n g b e i n g a p p r o x i m a t e l y i n a p l a n e a t r i g h t a n g l e st o t h e ax i s o f r o t a ti o n , th e n u m b e r o f p a r t ic l e s i n t h e r i n g sd i m i n i s h i n g a s t h e r a d i u s o f th e r i n g d i m i n is h e s. I f t h ec o r p u s c l e s c a n m o v e a t r i g h t a n g l e s t o t h e p l a n e o f t h e i ro rb i t , t he r ings w i ll be in d i f f e ren t p lanes ad jus t in g them se lvess o t h a t th e r e p u l si o n b e t w e e n t h e r i n g s i s b a la n c e d b y t h ea t t r ac t io n exer ted by the pos i t ive e lec t r i fi ca t ion o f the spherei n w h i c h t h e y a re p l a c e d . W e h a v e t h u s in th e fi rs t p l a c e asphere o f un i fo rm pos i t ive e lec t r i f i ca t ion , and in s ide th i ss p h e r e a n u m b e r o f c o r p u s c le s a r r a n g e d i n a s e ri e s o f p a r a l l e lr ings , the nu m ber o f co rpusc les in a r ing v a ry ing f rom rin~o-t o r i n g : e a c h c o r p u s c l e i s tr a v e l l i n g a t a h i g h s p e e d r o u n ~

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Structure of the Atom . 255the circumference of the r ing in which i t is s i tuated~ and therings are so arranged that those which contain a large n um berof corpuscles are near the surface of the spher% whi le thosein which there are a smaller num ber of corpuscles are morein the inside.If the corpuscles, l ike the poles of th e l i t t le mag nets inMayor 's experiments with the f loating magnets, are con-strained to move in one plane, they would, even if not inrotation, be in equil ibrium when arranged in the series ofrings jus t described. The rotation is required to ma ke thearrangem ent s table when the corpuscles can m ove at r igh tangles to the plane o f the r ing.

Application o f the preceding Results to the Theor~j of tl~eStructure of the Atom.W e suppose that the atom consists of a numb er of cor-puscles moving about in a sphere of uniform posit ive electr i-f ication : the problems we have to solve are (1) wh at wouldbe the structu re of such an atom~ i. e . how wo uld the cor-puscles arrange themselves in the sphere; and (2) wh atproperties wo uld this structu re c onfer upon the atom. Thesolution of (1) when the corpuscles are constrained to movein one plane is indica ted by the results we have ju s t o bta i ned --the corpu scles Will" arr ang e them selves in a series of conce ntricrings. This arrang em ent is necessitated by the fact that alarge number of corpuscles cannot be in stable equil ibriumwhen arranged as a single r ing, while this r ing can be madestable by placing inside i t an ap propriate nu m ber of cor-puscles. W he n the corpuscles are not constrained to oneplane, but can move about in all directions~ they will arrangethemselves in a series of concentric shells; for w e can easilysee that , as in the case of the r ing, a nm nber of corp usc les

distr ibuted over the surface of a shell wil l not be in stableequil ibrium if the number of corpuscles is large, unless thereare other corpus cles inside the shell, wh ile the equilibrim ncan be made stable by introducing within the shell an appro-priate number of other corl~useles.The analytical and geometrical diff icult ies of the problemof the distr ibution of the corpuscles when they are arrangedin shells are much great er than when the y are arrange d inrings, and I have not as yet succeeded in gett i ng a generalsolution. W e can so% how ever, that the same kind of pro-perties will be associated with the shells as w ith the rin.gs;and as our solution of the lat ter case enables us to givedefinite results, I shall confine myself to this case, andendeavour to show that the proper ties conferred on th eT 2

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256 Prof . J . J . Thomson on theatom by th is r ing s t ructure are analogous in many respectsto those po sses sedb y the a toms of the chemical e lements , andthat in part icular the properties of the atom will depend uponi ts a tomic weight in a wa y very analogous to tha t expressedby the per iodic law.Le t us suppose, then, th at w e have N corpuscles eachcar ryin g a charge e of negative electr ici ty, placed in a sp hereof posit ive electr if ication, the whole charge in the spherebein g equal to Ne; let us f ind the distr ibution of the cor-puscles whe n they are ar ranged in what we ma y consider tobe the s imples t way, i .e . when the num ber of r ings is aminimum, so that in each ring there are as nearly as possibleas many corpuscles as i t is possible for the corpuscles insideto hold in equil ibrium. Le t us suppose that th e num ber ofinternal corpuscles required to make the equil ibrium of aring of n corpuscles stable is f ( n ) . The value off (n) for aseries of va lues of n is give n in the table on pag e 254: ; inthat t ab le ' f ( n) i s denoted by p . The number of corpusclesin the ou ter r ing nl will then be determined by the condit ionthat N -- n l , the num ber of corpuscles ins ide , mast be jus tsufficient to keep the r ing of n 1 corpuscles in equil ibrium,i. e. , n I wil l be determined by the equat ion

N - - h i = f (n ,) . . . . . . . (1)I f the value of n l got f rom this equat ion is not an in tegerwe must take the in tegra l par t of the value .To get n~, the number of corpuscles in the second ring, wenotice that there must be I~ --n l--n ~ corpuscles inside; hencen~ is given by the equation

N - nl - - n~ = f (n2 ) .Similarly, n~, n~, . . . , the nmnber of corpuscles in the 3rd,

4th, &c. r ings reckoned from the outside, are given byI ~ - ~ - , ~ - ~ = f (~),1 ~ - . 1 - ~ 2 - ~ - , ~ = (n~).

These equations can be solved very rapidly by a graphicalmethod. Draw the graph whose abscissa = f ( n ) and whoseordinate is n. The values of f ( n ) for a series of values of nare given on page 25]: ; from these values the curve f ig. 1has been constructed.To find how a num ber of corpuscles equal to N will arrangethem selves, me asure off on the axis o f abseissm a distanc efrom O equal to lq. Le t OP be this distance, throu gh Pdra w I~Q inclined at an angle of 135 ~ to the horizontal axis,

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Structure of the Atom. 257cut t ing the curve in Q, draw the ordinate QM; then theintegral part of QM will be the vah~e of nl, the nmnb er of

Fig. 1.fJ

Mez0 MjPz 40 ~ 60 P ~0corpuscles in the f irst r ing reckoned from the outside. Fo revident ly =/(Q),a n d 0 M = 0 N - - N M , a n d s in ce P Q is i nc li ne d at 45 ~ t o t h eaxis , NM = 0 M ; hence

0 N - - Q M = f ( Q ~ I ) .Com paring th is wi th equat ion ( i ) we see tha t the in teg ra lpa rt of Q M is the valu e of" nl.

To get the value of n~, the nu m ber of corpuscles in thesecond ring, we mark offthe abscissa OP1----N--nl ( if QM isan integ er P, w ill coincide with M), then from P1 draw1)~Q~ parallel to P Q cutt i ng the cu rve in Q1; the integralpa rt os Q1MI will be the value of n~. To get n3 m ark off theabscissa OP~=N--nl--n2, and draw P2Q 2 parallel to PQ ; th eintegral part of Q2M~ will be the v alue of n3. In this w ay wecan in a very short t ime find the configuration.The following table, ahich gives the way in which variousnum bers of corpuscles group themselves, has been calculatedin th is w ay ; the numbers range downw ards f rom 60 a tintervals of 5.fNu mb er of corpuscles . . . . . . . . . 60. | 55. 50. 45. 40. 35.

N um bo n 2 --;- - T g - , - S ? - - - Y (16 | 14; [ 15 [ 14 ] 13 [ 1213 | 12 11 10 8 ~il ~ 'i I 5 I 4 I 3 I 1Nu mb er of 30. 25. 20. 15. 10. 5.Nu-mbe--r in successive r ings . . . - - ~ - - I - - ~ - o ' [- 12 -- 1 10 ~ 8 [ 510 | 9 | 7 I 5 ] 2 /

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2 5 8 P r o f . J . J . T h o m s o n on theW e g i v e a l s o t h e e n t ir e s e ri e s o f a r r a n g e m e n t o f c o r p u s c l e sfo r wh ich the ou te r r ing cons i s t s o f 20 co rpusc les .

~ u m b e r o f c o r p n s e l e s . . . . . . . . . 5 9 . 6 0 . 6 1 . 6 2 . 6 3 . 6 4 . 6 5 . 6 6 . 6 7 . I/: N u m b e r i n s u e c o s s i v e r i n g s . . . l 2 0 I 2 0 } 2 0 I 2 0 I 2 0 I 2 0 / 2 0 I 2 0 f 2 0 // 1 6 I 1 6 / 1 6 I 1 7 I 1 7 I 1 7 / 1 7 I 1 7 4 1 7 /I 1 3 I l ~ / 1 3 / 1 3 / 1 3 I l ~ / 1 4 I l ~ / 1 5 // 8 1 a / 9 / 9 1 ~ o l l o / ~ o i l o / l o /

5 9 i s t h e s m a l l e st n u m b e r o f c o r p u s c l e s w h i c h c a n h a v e a no u t e r r i n g o f 2 0 , w h i le w h e n t h e n u m b e r o f c o rp u s c l e s i sg r e a t e r t h a n 6 7 t h e o u t e r r i n g w i l l c o n t a i n m o r e t h a n 2 0corpusc les .L e t u s n o w c o n si d e r t h e c o n n e x i o n b e t w e e n t h e s e r e s u l t sa n d t h e p r o p e r t i e s p o ss e ss e d b y t h e a t o m s o f t h e c h e m i c a le l e m e n t s . W e su p p o s e t h a t t h e m a s s o f a n a t o m i s t h e s t u nof" the m asses o f the co rpusc les i t con ta ins , so tha t the a tom icw e i g h t o f a n e l e m e n t i s m e a s u r e d b y t h e n u m b e r o f c o r -p u s c l e s i n it s a t o m . " A n i n s p e c t io n o f t h e r e s u l ts j u s t g i v e nw i l l s h o w t h a t s y s te m s b u i l t u p o f r i n g s o f c o r p u s c le s i n t h ew ay we ha ve desc r ibed , w i l l pos ses s p rope r t i e s a na logo us tos o m e o f t h o s e p o s se s se d b y t h e a t o m . I n t h e f i rs t p l a c e , w es e e t h a t t h e v a r i o u s a r r a n g e m e n t s o f t h e c o r p u s c le s c a n b ec l a ss if ie d i n f a m i li e s, t h e g r o u p i n g o f t h e c o r p u s c l e s i n t h ev a r i o u s m e m b e r s o f t h e f a m i l y h a v i n g c e r t a in f e a t u r e s i nc o m m o n . T h u s , f o r e x a m p l e , w e s ee t h a t t h e g r o u p o f 6 0c o r p u s c l e s c o n s is ts o f t h e s a m e r i n g s o f c o r p u s c l e s a s t h eg ro up o f 40 w i th an ad d i t iona l r ing o f "20 co rpusc les ro un d i t ,w h i l e t h e g r o u p o f 4 0 c o n si st s o f t h e s a m e s e r ie s o f r i n g s a st h e g r o u p o f 2 4 w i t h a n a d d i t io n a l r i n g o u t s id e , w h i l e 2 4 i st h e g r o u p 1 1 w i t h a n a d d i t i o n a l r i n g . T o c o n t i n u e t h es e ri es f o r l a r g e r n u m b e r s o f c o rp u s cl es , t a k e t h e c u r v e x = f ( y )w h e n f ( n ) i s t h e n m n b e r o f c o r p u s c l e s t h a t m u s t b e p l a c e d

Fire 2.

\ \ \, o / -inside a r i n g o f ~ c o r p u s c le s to m a k e i t s t a b le . L e t Q b ethe po in t on tb i s cu rve co r resp ond ing to 60 co rpusc les , i . e .O P = 6 0 , f r o m Q d r a w Q P 1 i n c l in e d a t a n a n g l e o f 1 3 5 ~ t o

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Struc ture o f the A tom . 259t h e a x i s o f x ; t h e n t h e n m n b e r o f c o r p u s c l e s r e p r e s e n t e d b yO P 1 w i l l b e a r r a n g e d l i k e t h e 6 0 c o r p u s c l e s w i t h a n a d d i t i o nr ing o f Q1P1 co rpusc les ( f ig . 2 ) . To f ind the ne x t m em ber o f thefam i ly , d raw Q1P2 para l l e l to QP1 cu t t ing the ax i s o f x in P~ ,t h e n O P ~ w i l l r e p r e s e n t t h e n u m b e r o f c o r p u s c l e s i n t h e n e x tm e m b e r o f t h e f a m i l y ; a n d b y c o n t i n u in g t h e p ro c e ss w ec a n f i n d t h e s u c c e s si v e m e m b e r s . T h u s w e s e e t h a t w e c a nd i v i d e t h e v a r io u s g r o u p s o f a t o m s i n t o s e r ie s s u c h t h a t e a c hm e m b e r o f t h e se r ie s i s d e r i v e d f r o m t h e p r e c e d i n g m e m b e r( i . e . t h e m e m b e r n e x t b e l o w i t i n a t o m i c w e i g h t ) b y a d d i n gt o i t a n o t h e r r i n g o f' c o r p u s c le s . W e s h o u l d e x p e c t t h ea t o m s f o r m e d b y a s e r ie s o f c o r p u s c l e s o f t h i s k i n d t o h a v em a n y p o i n t s o f r e s e m b l a n c e . T a k e , f o r e x a m p l e , t h e v i b r a -t i o ns o f t h e c o r p u s c l e s ; t h e s e m a y b e d i v i d e d i n to t w os e ts : - - ( 1 ) T h o s e a r i s i n g f r o m t h e r o t a t i o n o f t h e c o r p u sc l e sa ro un d the i r o rb i t s : i f a l l the co rpusc les in one a tom have thes a m e a n g u l a r v e l o c i t y , t h e f r e q u e n c y o f t h e v i b r a t i o n s p r o -d u c e d b y th e r o t a t io n o f t h e r i n g o f c o r p u s cl e s is p r o p o r t i o n a lt o t h e n u m b e r o f c o r p u s c l e s in t h e r i n g ; a n d t h u s i n t h es p e c t r u m o f e a c h e l e m e n t i n t h e s e ri e s th e r e w o u l d b e a se r ie so f f r e q u e n c i e s b e a r i n g t h e s a m e r a t i o t o e a c h o t h e r , t h e r a t i oo f t h e f r e q u e n c ie s b e i n g t h e r a ti o s o f t h e n u m b e r s i n t h evar ious r ings .The s econd sys tem o f' v ib ra t ions a re those a r i s ing f rom thed i s p l a c e m e n t o f t h e r i n g f r o m it s c i r c u l a r f i g u r e . I f n o w t h ed i s ta n c e o f a c o r p u s c le i n t h e o u t e r r i n g f r o m a c o r p u s c l e i nthe co l l ec t ion o f' r ings in s ide i t i s g re a t com pared w i th thed i s t a n c e o f t h e s e c o n d c o r p u s c l e f r o m i ts n e a r e s t n e i g h b o u ron i t s ow n r ing , the e f f ec t o f the ou te r r ing of" co rpusc les onthe inner s e t o f r in gs w i l l on ly " d i s t u r b " t h e v i b r a t i o n s o ft h e l a t t e r w i t h o u t f u n d a m e n t a l l y a l t e r i n g t h e c h a r a c t e r o tthe i r v ib ra t ions . Thu s fo r these v ib ra t ions , a s we l l a s fo rt h o s e d u e t o th e r o t a ti o n s , t h e s e q u e n c e o f f r e q u e n c ie s w o u l dp r e s e n t m u c h t h e s a m e f e a t u r e s f o r t h e v a r i o u s e l e m e n t s i nt h e s e r i e s ; t h e r e w o u l d b e in t h e s p e c t r u m c o r r e s p o n d i n ggro up s o f a s soc ia ted l ines . W e rega rd a se r i e s o f a tomsf o r m e d i n t h i s w a y , i. e. w h e n t h e a t o m o f t h e p t h m e m b e r i sf o r m e d f r o m t h a t o f t h e ( p - - 1 ) t h b y t h e a d d i ti o n o f a s in g l er i n g o f c o r p u sc l e s, a s b e l o n g i n g t o e l e m e n t s in t h e s a m e g r o u pi n t h e a r r a n g e m e n t o f t h e e l e m e n t s a c c o r d i n g t o t h e p e r i o d iclaw ; i . e . , they fo rm a s e r i e s wh ich , i f a r r anged accord ing toM e n d e l 6 e f ' s t a b l e , w o u l d a l l b e i n t h e s a m e v e r t i c a l c o l u m n .T h e g r a d u a l c h a n g e i n t h e p r o p e r t ie s o f th e e l e m e n t s w h i c ht a k e s p la c e a s w e t r a v e l a l o n g o n e o f t h e h o r i z o n ta l r o w s i nM e n d e l d e f ' s a r r a n g e m e n t o f t h e e l e m e n t s, i s a ls o i l lu s t ra t e d b yt h e p r o p e r t i e s p o s s es s e d b y t h e s e g r o u p s o f c o r p u s c le s . T h u s

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2 6 0 P r o f . J . J . T h o m s o n o n thec o n s id e r t h e s e ri e s o f a r r a n g e m e n t s o f t h e c o r p u s c l e s g i v e n o np . 258 , in a l l o f wh ich the ou te r r ing con ta ins 20 co rpusc lesA n o u t e r r o w o f 2 0 c o r p u s cl e s f i rs t o c c u r s w i t h 5 9 c o r p u s c l e s ;i n t h i s c a s e t h e n u m b e r o f c o r p u s c l e s i n s id e i s o n l y j u s tsu f fi c ien t to m ake the ou te r r ing s t ab le ; th i s r in g w i l l the re -f o r e b e o n t h e v e r g e o f i n s t a b il i ty , a n d w h e n t h e c o r p u s c l e si n t h i s r i n g a r e d i sp l a c e d t h e f o r c e s o f r e s t i tu t i o n u r g i n gt h e m b a c k to t h e i r o r i g in a l p o s i t i o n w i l l b e sm a l l. T h u s w h e nt h i s r i n g i s s u b j e c t e d to d i s t u r b a n c e s f ro m a n e x t e r n a l s o u r c e ,o n e o r m o r e c o r p u s c le s m a y e a s i ly b e d e t a c h e d f r o m it ; s u c hau a tom the r e fo r e w i l l eas i ly lo se a ne ga t iv e ly e lec t r i fi ed co r -p u s c l e , a n d t h u s a c q u i r e a c h a r g e o f p o s i t i v e e l e c t r i c i t y ;s u c h a n a t o m w o u l d b e h a v e l i k e t h e a t o m o f a s t r o n g l ye l e c t ro p o s i t iv e e l e m e n t. W h e n w e p a s s f r o m 5 9 t o 6 0 c o r -p u s c l e s t h e o u t e r r i n g i s m o r e s t a b le , b e c a u s e t h e r e i sa u a d d i t i o n a l c o r p u s c l e i n s id e i t ; t h e c o r r e s p o n d i n g a t o mw i l l t h u s n o t b e s o e l e c tr o p o s i ti v e a s t h a t c o n t a i n in g o n l y5 9 c o r p u s c l e s . T h e a d d i ti o n o f e a c h s u c c e s si v e c o r p u s c l ewi l l m ake i t m ore d i f f icu l t to de tach co rpusc les f rom theo u t e r r i n g , a n d w i l l t h e r e f o r e m a k e t h e a t o m l es s e l e c t r o -p o s i ti v e . W h e n th e s ta b i l i t y o f t h e o u t e r r i n g g e t s v e r y g r ea t ~i t m a y b e p o s s ib l e f o r o n e o r m o r e c o r p u s c le s t o b e o n t h es u r f a c e o f t h e a t o m w i t h o u t b r e a k i n g u p t h e r i n g ; i n th i sc a se t h e a t o m c o u l d r e c e i v e a c h a r g e o f n e g a t i v e e l e c t r i c i ty ,a n d w o u l d b e h a v e l ik e t h e a t o m o f a n e l e c t ro n e g a t i v e e l e m e n t .T h e i n c r e a s e i n t h e s t a b i l i ty o f t h e r i n g , a n d c o n s e q u e n t l y int h e e l e c t r o n e g a t i v e c h a r a c t e r o f th e a t o m , w o u l d g o o ai n c r e a s i n g u n t i l w e h a d a s m a n y a s 6 7 c o r p u s c le s , w h e n t h es t a b i l i ty o f t h e o u t e r r i n g w o u l d b e a t a m a x i m u m . A g r e a tc h a n g e i n t h e p r o p e r t i e s o f t h e a t o m w o u h l o c c u r w i t h 6 8c o r p u s c l e s , f o r n o w t h e n u m b e r o f c o r p u s c l e s i n t h e o u t e rr ing inc reases to 21 ; these 21 co rpusc les a r e~ how ever , on lyj u s t s ta b l e , a n d w o u l d , li k e t h e o u t e r r i n g o f 2 0 i n t h ea r r a n g e m e n t o f t h e 5 9 c o r p u s c l e s, r e a d i l y l os e a c o r p u s c l ea n d s o m a k e t h e a t o m s t r o n g l y e l e c t r o p o s i t i v e .T h e p r o p e r t i e s o f t h e g r o u p s o f 5 9 a n d 6 7 c o r p u sc l e s, w h i c ha r e r e s p e c t i v e l y a t t h e b e g i n n i n g a n d e n d o f t h e s e r i e s w h i c h h a san ou te r r ing o f 20 co rpusc les , dese rv e espec ia l cons ide ra t ion .T h e a r r a n g e m e n t o f c o r p u s c l e s i n t h e g r o u p o f 5 9 , a l t h o u g hv e r y n e a r t h e v e r g e o f i n s ta b i l it y , a n d t h e r e f o r e v e r y l i a b l eto lo se a co rpusc le and the reb y acq u i r e a pos i t ive charge~w o u l d n o t b e a b l e t o r e t a i n t h is c h a r g e . F o r w h e n i th a d l o s t a c o r p u s c le , th e 5 8 c o r p u s c l e s l e ft w o u l d a r r a n g et h e m s e l v e s i n t h e g r o u p i n g c o r r e s p o n d i n g t o 5 8 c o r p u s c l e sw hich i s the l a s t to have an ou te r r ing o f 19 co rpusc les ; th i sr ing i s the r e fo re exce ed ing ly s t ab le so tha t no fu r the r cor --

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Str,ctuTe of the Atom. 261puscles wou ld escape fro m it , while the positive charge onthe system due to the escape of the 59th corpuscle wouldattract the surro undi ng corpuscles. Thus this arran gem entcould not remain perm anen tly charged ; for as soon as onecorpuscle had escaped it would be replaced by another. Anatom constituted in this way would be neither electropositivenor electronegative, hut one incapable of receiving perm anent lya charge of electricity.The group containing 60 corpuscles would be the mostclectropositive of the series ; but this could only lose onecorpuscle; i . e. acquire a charge of one unit of positiveelectricity; for if i t lost two we should hav e 58 corpu scl es- -as when the group of 59 had lost one cor pus cle --a nd in thiscase the system would be even more l ikely than the other toattract external corpuscles, for it would have a charge of tw ounits of positive electricity instead of one. Thus the syst emcontaining 60 corpuscles would get charged with one, butonty one, unit of positive elec tric ity: i t wou ld there fore actl ike the atom of a monovalent electropositive elem ent.The group containing 61 corpuscles would not part withits corpuscles so readily as the g roup of 60, bu t on the otherhand it could afford to lose two, as it is not until it has lostthree that i ts corpuscles are reduced to 58~ when, as we haveseen, i t begins to acquire fresh corpuscles. Thus this systemnfight get charg ed w ith two units of positive electricity, andwould act l ike the atom of a divalent electropositive element.Similarly t h e grou p of 62, thou gh less l iable even thanthe 61 to lose its corpuscles, could, on the other hand, lose 3without beginning to recover i ts corpuscles ; i t could thusacquire a charge of 3 units of positive electricity, and wouldact l ike the atom of a tr ivalent electropositive element.Let us now go to the groups at the other end of the seriesand consider the pro perti es of the last of' the series, the g rou pof 67 corpuscles. The outer r ing would be ver y stable, butif the system acqu ired another corpuscle, the 68 corpuscleswould arrang e themse lves with a r ing of' 21 corpuscles on theoutside ; as 68 is the smallest number of corpuscles with anouter r ing of 21 , the r ing is very near ly unstable and eas i lyloses a corpus cle. Th us the group o[ 67 corpuscles~ as soon asit acquires a n egativ e charge, would lose i t again, and thesystem, l ike the grou p of 59, would be incapable of beingperman ent ly charged with e lec tr ic i t y-- i t would ac t l ike theatom of an element of no valency.

The group of 66 would be the most e lec tronegat ive of theseries, but this wou ld only be able to retain a charge o f oneuni t of neg ativ e elec trici ty ; for if" it acqui red 2 units ther e

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2 6 2 P r o f . J . J . T h o m s o u o ~ t ] ~ ew o u l d b e 6 8 c o rp u s cl es , a n a r r a n g e m e n t w h i c h , a s w e h a v es e e n , r a p i d l y l os e s i ts c o r p u s c le s . T h i s g r o u p o f 6 6 w o u l dt h e r e f o r e a c t l i k e t h e a t o m o f' a m o n o v a l e n t e l e c t r o n e g a t i v ee l e m e n t .T h e g r o u p o f 6 5 w o u l d b e l e s s li a b l e t h a n t h a t o f 6 6 t oa c q u i r e n e g a t i v e c o r p u sc l es , b u t , o n th e o t h e r h a n d , it w o u l du n d e r s u i ta b l e c i r c u m s t a n c e s b e a b l e t o r e t a i n 2 c o r p u s c le s ,a n d t h u s b e c h a r g e d w i t h 2 u n i t s o f n e g a t i v e e l e c t r i c i t y , a n dw o u l d a c t l i k e t h e a t o m o f a d i v a l e n t e l e c t r o n e g a t i v e e l e m e n t .S i m i l a r l y , t h e g r o u p o f 6 4 w o u l d a c t l i k e t h e a t o m o f at r i v a l e n t e l e c t r o n e g a t i v e e l e m e n t , a n d s o o n .T h u s , i f w e c o n s i d e r t h e s e r ie s o f' a r r a n g e m e n t s o f c o r p u s c l e sh a v i n g o n th e o u t si de a r i n g c o n t a i n i n g a c o n s t a n t n u m b e ro f c o rp u s c le s , w e h a v e a t t h e b e g i n n i n g a n d e n d s y s t e m sw h i c h b e h a v e l i k e t h e a t o m s o f a n e l e m e n t w h o se a t o m s a r ei n c a p a b l e o f r e t a i n i n g a c h a r g e o f e i th e r p o s i ti v e o r n e g a t i v ee l e c t r i c i t y ; t h e n ( p r o c e e d i n g i n t h e o r d e r o f i n c r e a s i n gn u m b e r o f c o r p u s cl e s) w e h a v e f ir s t a s y s te m w h i c h b e h a v e sl ik e t h e a t o m o f a m o n o v a l e n t e le c t r o p o s it iv e e l e m e n t , n e x to n e w h i c h b e h a v e s l ik e t h e a t o m o f a d i v a l e n t e l e e t ro p o s i t iv ee le ~m e nt, w h i l e a t t h e o t h e r e n d o f t h e s e r ie s w e h a v e a s y s t e mw h i c h b e h a v e s l i k e a n a t o m w i t h n o ~ , al en c y, i m m e d i a t e l yp r e c e d i n g t h is , o n e w h i c h b e h a v e s l i k e t h e a to m o f a m o n o -v a l e n t e l e c t r o n e g a t i v e e l e m e n t , w h i l e t h is a g a i n i s p r e c e d e db y o n e b e h a v i n g l i k e t h e a t o m o f a d i v a l e n t e l ec tr o n e ga ti _ v ec l e m e n t .T h i s s e q u e n c e o f p r o p e r t i e s i s v e r y l i k e t h a t o b s er v e d i nt h e c a s e o f t h e a t o m s o f t h e e l e m e n t s .T h u s w e h a v e t h e s e r i e s o f e l e m e n t s :

H e L i B e B C .N O F N e .:N e ~ a M g A 1 S i P S C1 A r g .T h e f i r s t a n d l a s t e l e m e n t i n e a c h o f t h e s e s e ri e s h a s n ov a l e n c y , t h e s e co n d is a m o n o v a l e n t e l e c t r o p o s it iv e e l e m e n t ,t h e l a s t b u t o n e is a m o n o v a l e n t e l e c t r o n e g a t iv e e l e m e n t , t h et h i r d i s a d i v a l e n t e le c t r o p o s i t iv e e l e m e n t , ' t h e l a s t b u t t w o ad i v a l e n t e l e c t r o n e g a t iv e e l e m e n t , a n d s o o n .W h e n a t o m s l i k e th e e l e c t r o n e g a t i v e o n es , i n w h i c h t h ec o r p u s c l e s a r e v e r y s t a b l e , a r e m i x e d w i th a t o m s l i k e t h ee l e c t ro p o s i ti v e o ne s~ i n w h i c h t h e c o r p u s c le s a r e n o t n e a r l ys o f i r m l y h e l d , t h e f o r c e s t o w h i c h t h e c o r p u s c l e s a r e s u b j e c tb y t h e a c ti o n o f t h e a t o m s u p o n e a c h o t h e r m a y r e s u lt i n th ed e t a c h m e n t o f c o r p u s cl e s " f r o m t h e e l e c tr o p o s i ti v e a t o m s

a n d th e i r t ra n s f e r e n c e t o th e e l e c t ro n e g a t i v e . T h e e l e c tr o -n e g a t i v e a t o m s x~ l l t h u s g e t a c h a r g e o f n e g a t i v e e l e c t r i c i ty ,t h e e l e c t r o p o s i t i v e a t o m s o n e o f p o s i t i v e , t h e o p p o s i t e l yc h a r g e d a t o m s w i l l a t t r a c t e a c h o t h e r , a n d a c h e m i c a l

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Structure o f t]te A tom . 263compound of the electropositive and electronegative atomswill be formed.Just as an uncharged conducting sphere will by electro-static induction attract a corpuscle in its neighbourhood, soa corpuscle outside an atom will be attracted, even thoughthe atom has not become positively charged by losing acorpuscle. When the outside corpuscle is dragged into theatom there will be a diminution in the potential energy,the amount of this diminution depending on the number ofcorpuscles in the atom. If now we have an atom A suchthat loss of potential energy due to the fall into the atom ofa corpuscle from outside is greater than the work required todrag a corpuscle from an atom B of a different kind, then anintimate mixture of-A and B atoms will result in the A atomsdragging corpuscles from the B atoms, thus the A atoms willget negatively, the B atoms positively electrified, and theoppositely electrified atoms will combine, fbrming a com-pound such as A_ B+ ; in such a case as this chemicalcombination might be expected whenever the atoms werebrought into contact. Even when the loss of potentialenergy when a corpuscle falls into A is less than the workrequired to drag a corpuscle right away from B, the existenceof a suitable physical environment may lead to chemicalcombination between A and B. For when a corpuscleis dragged out of and away from an atom a considerableportion of the work is spent on the corpuscle after it has lef~the atom, while of the work gained when a corpuscle fallsinto an atom, the proportion done outside to that done insidethe atom is smaller than the proportion for the correspondingquantities when the corpuscle is dragged out of an atom.Thus, though the work required to move a corpuscle from Bto an infinite distance may be greater than that gained whena corpuscle moves from an infinite distance into A, yet thework gained when a corpuscle went from the surface of Ainto its interior might be greater than the work required tomove a corpuscle from the interior to the surface of B. inthis case anything which diminished the forces on thecorpuscle when they got outside the atom, as, for exampl%the presence of a medium of great specific inductive capacitysuch as water, or contact with a metal such as platinumblack, would grea tly increase the chance of chemical com-bination.

The Existence o f Secondary Groups o f Corpuscleswith in the Atom.The expression given on p. 238 for the radius of a ring ofcorpuscles shows that it depends on ue/b , where re is the

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26~ l~ro f . J . J . Thomson on t /~eam ou n t o f pos i t ive e lec tr i f ica t ion w i th in a sphere o f r ad ius b :47 rt h u s v e / b ~ i s e q u a l t o - ~ 3 p ' w h e r e p i s t h e d e n s i t y o f t h epos i t ive e lec t r if i ca t ion in the sphere : thus , i f the den s i ty o fthe e lec t r i f i ca t ion be kep t cons tan t , the r ad ius o f the r ing w i l lb e i n d e p e n d e n t o f t h e s iz e o f t h e s p h e re . N o w l e t u s t a k e ala rge sphere and p lace w i th in i t a r ing o f such a s ize tha tt h e r i n g w o u l d b e i n st a b le e q u i l i b r i u m i f it s c e n t r e w e r ea t the cen t r e o f the sphere . To f ix ou r ideas, l e t u s t ake thecase o f th ree co rpusc les a t the co rner s o f an equ i la te ra lt r i ang le , and p lace th i s t r i ang le so tha t i t s cen t r e 0 ~ i s nolong er a t the cen t r e o f the sphere : we can e as i ly s ee ~hat ~hecorpu sc les w i l l r emain a t the co rner s o f an equ i la te ra l t r i ang leo f the s am e s ize , and tha t the t r i ang le w i l l m ove l ike a r ig idb o d y a c t e d u p o n b y a f o r c e p r o p o r t i o n a l to t h e d i s ta n c e o fi ts c e n t r e f r o m 0 t h e c e n t r e o t t h e s p h e r e. T o p r o v e t h i sw e n o t i c e t h a t t h e r e p u l s io n b e t w e e n t h e c o r p u s c i e s i s t h es a m e a s w h e n t h e c e n t r e o f t h e t r i a n g l e i s a t 0 . T h e a t t r a c ti o no f t h e s p h e r e o n a c o r p u s c le P i s p r o p o r t i o n a l t o O F , a n d s omay be r eso lved in to two fo rces , one p ropor t iona l to O~Pa l o n g P O ~ ( 0 ' i s t h e c e n t r e o f ti m t r i a n g l e ) a n d t h e o t h e rp r o p o r ti o n a l t o O 0 ~ a c t i n g a l o n g 0 ~ 0 . N o w t h e c o r p u sc l e sa r e b y h y p o t h e s i s i n e q u i l i b r iu m u n d e r t h e i r m u t u a l r e -pu l s ions , and the a t t r ac t ion to the ce n t r e p ropo r t iona l to O~P:t h u s t h e r e l a t i v e p o s i t i o n o f t h e c o r p u s c l e s w i l l r e m a i nu n a l t e r e d , a n d t h e s y s t e m o f t h r e e c o r p u sc l e s w i ll m o v e a sa r i g i d b o d y u n d e r a c e n t r a l f o r c e a c t i n g o n i ts c e n t r e o fg r a v i t y p r o p o r t i o n a l t o t h e d i s ta n c e o f t h a t p o i n t f r o m t h ecen t re o t' t he sphere .The th ree co rpusc les w i l l , a t a po in t whose d i s tance f romt h e i r c e n t r e i s la r g e c o m p a r e d w i t h a s i d e o f t h e t ri a n g l e ,p r o d u c e t h e s a m e e f f e c t a s i f t h e c h a r g e s o n t h e t h r e ec o r p u s c l e s w e r e c o n d e n s e d a t t h e c e n t r e o f t h e t r i a n g l e ; t h e ywi l l th us a t such po in t s ac~ ]ike a un i t , an d the r esu l t s w eh a v e p r e v i o u s l y o b t a in e d f o r s i n g le c o r p u sc l e s m a y b e e x -t e n d e d t o t h e c a s e w h e n t h e s i n g le c o r p u s c le s a r e r e p la c e db y r i n g s o f c o r p u s c l e s w h i c h w o u l d b y t h e m s e l v e s b e i ne q u i l i b r i u m . I t s h o u l d h e n o t e d t h u t t h e a t o m in w h i c ht h e s e s y s t e m s a r e p l a c e d m u s t b e l a r g e e n o u g h t o a l l o w t h e s er i n g s o f' c o r p u s c l e s - - s u b - a t o m s w e m a y c a ll t h e m , t o b es e p a r a t e d b y d i s t a n c e s c o n s i d e r a b l y g r e a t e r t h a n t h e d i s t a n c eb e t w e e n t h e c o r p u s c le s i n o n e o f t h e r in g s .I f w e r e g a r d t h e a t o m s o f t h e h e n . l e t e l em e n t s a s p r o d u c e db y the coa lescence o f l igh te r a toms , i t i s r easonab le to supp oset h a t t h e c o r p u s c le s i n t h e h e a v i e r a t o m s m a y b e a r r a n g e d i ns e c o n d a r y g r o u p s o r s u b - a t o m s , e a c h o f t h e se g r o u p s a c t i n g

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Structure of the Atom. 265as a unit . W he n the corpuscles are done up in bundles inthis way , i t is possible to have stabil i ty whe n these bundlesare arran ged in a r ing wi~h a smaller nu m ber of corpusclesinside than whe n the corpuscles in the bundles are a rrange dat equal in tervals round the c i rcumference of the r ing.Thus, take the case of a r ing of 30 corpuscles; if thesewe re arrange d at equal intervals, 101 corpuscles would berequired inside the r ing to make i t s table. If , however, the30 corpuscles were group ed in ten sets of three each, only3 x 3 = 9 corpuscles in the in terior would be required tomake the ar rangement s table .

Constitution of the Atom of a Radioactive Element.Our study of the stabil i ty of systems of corpuscles hasmade us acquainted with systems which are stable when thecorpuscles are rotating with an angular velocity greater thana certain value, but which become unstable when the velocityfalls below this va lue. Thus, to take an instance~ we saw(p. "249) tha t fou r corpusc les can b e stable in one plane atthe corners of a square, if they are rotating with aa angularveloci ty greater than "325ve'~/mb~, but become unstable if the

velocity fal ls below this velocity, the corpuscles in this casetending to place themselves at the corners of a tetrahedron.Consider now the properties of an atom containing a systemof' corpuscles of this k ind, suppose the corpus cles wereorigiHally moving with velocit ies far exceeding the cri t icalveloci ty ; in consequence of the radia t ion f rom the movingcorpuscles, the i r ve loci ties wi ll s lo w ly-- ver y s low ly--dimin ish;when, after a long interval, the veloc ity reaches the cri ticalvelocity~ the re will b e w ha t is equiv alent to an explosion ofthe corpuscles, the corpuscles will mov e far aw ay from theiroriginal positions, their potential energy will decrease, whiletheir kinetic en erg y will increase. The kinetic ene rgygained in this w ay mig ht be sufficient to ca rry the systemout of the atom , and w e should hav% as in the case of radium,a part of the atom shot off. In consequ ence of the very slowdissipation of energy by radiation the l ife of the atom wouldbe very long. W e have taken the case of the four corpusclesas the typ e of a system which, l ike a top, requires for i tsstabil i ty a certain am oun t of rotation. A ny system possessingthis property would, in consequence of the gradual dissi-pation of en erg y by. radiation, give to the atom containin~,i t radioactive properhes similar to those conferred by the fourcorpuscles.

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