synopsis of ele men 00 carr rich
DESCRIPTION
The Genius of RamanujanTRANSCRIPT
^/^.
\
:
A SYNOPSIS
ELEMENTAIIY RESULTS
PUKE MATHEMATICSCONTAINTNO
PROPOSTTTONS.'FORMUL^, AND METHODS OF ANALYSIS.WITH
ABRIDGED DEMONSTRATIONS.SlTPLEMFNTFI)BEItY AN InDEX TO THE PaI'ERS ON Pi HE JIaTHEMATU S -WHUJI AKE TO FOVNn IN THE PHINCIPAL JoiBNALS AND TllANSACTlONS OF LEAUM I) Socill lEP, poTH English and Foreion, of the present centuky.
G.
S.
CARR, M.A.
LONDON
FRANCIS noi)(;soN,
farkin(tDon street, e.c CAMBRIDGE: MACMILLAN & BOWES.801886.(AUrights reserved)
:
C3neenngLibrai:'^
LONDON
PRINTED BY
C.
F.
HODGSON AND
SON,
GOUGH SQUARE, FLEET STREET.
76.
6j6
N TVERSJT?
PREFACE TO PART
I
Tin: work, of which
tlio
part
now
issued
is
a
first instal-
ment, has been compiled from notes made at various periodsof the last fourteen years, of teaching.
and
chiefly during* the
engagements
!Many of the abbreviated methods and mnemonic
rules are in thepupils.
form in which
I originally
wrote them for
my
The general object
of
the compilation
is,
as
the
title
indicates, to present within a moderate
compass the funda-
mental theorems, formulas, and processes in the chief branchesof pureTlie
and applied mathematics.
work
is
intended, in thetl,ic
first
place,
to follow
andit
supplement the use ofarrangedwitli tlie
ordinary text-books, andtlie
is
view of assisting
student in the task of
revision of book-w^ork.
To
this
end
I have, in
many
cases,
merely indicated the salient points of a demonstration, or merely referred to the theorems by which the propositionproved.Iis
am
convinced that
it
is
more
beneficial to tlie
student to recall demonstrations with such aids, than to read
and re-read them.
Let them be read once, but recalled often.
The
difference in the effect
upon the mind between reading a
mathematical demonstration, and originating cue wholly or
IV
PEEFACE.
partly,
is
very great.
It
may be compared
to tlic difference
between the pleasure experienced, and interest aroused, whenin the one case a travelleris
passively conducted through the
roads of a novel and unexplored country, and in the othercase he discovers the roads for himself with the assistance of
a map.
In the second place, I venture to hope that the work,
when completed, may provean aide-memoire and book
useful to advanced students as
of reference.
The boundary
of
mathematical science forms, year by year, an ever wideningcircle,
and the advantage
of
having at hand some condensed
statement of results becomes more and more evident.
To
the original investigator occupied with abstruse re-
searches in some one of the
many branches
of mathematics, a
work which gathers togethersitions in all,
synoptically the leading propo-
may not
therefore prove unacceptable.
Abler
hands than mine undoubtedly, might have undertaken the taskof
making such a digest
;
but abler hands might
also,
perhaps,
be more
usefully emj^loycd,
and with
this reflection I
have the
less hesitation in
commencing the work myself.is
The designin
which I have indicatedrelation toit
somewhat comprehensive, and
the present essay
may be regardeditit
as tentative.
The degree of success whichsuggestions or criticisms whichIc:!
may meetmay
with, and the
call forth, will
doubt-
have their
effect
on the subsequent portions of the work.
With
respect to the abridgment of the demonstrations, Ithat while
may remark,
some diffuseness
of explanation is not
only allowable but very desirable in an initiatory treatise,concisenessis
one of the chief reciuiremcnts in a work intended
PREFACE.
VIn order,
for
tlio
piii-i)OSos
of revision
and rcfiTeiico only.
liowever, not to sacrifice clearness to conciseness,la])our has
much moroThe onlyis
been expended upon this part of the subject-matterwill at first sip^ht
of the
book than
be at
all
evident.
])alpal)le I'esult lK'in
U
CONTENTS.
XIN... 1...
c" is...
incommensurable...
... .
...
...
21*52'JG'^^^2
AnnuitiesPROr.AniLITIES......
...
^*^^
Inequalities
...
...
... ... Arithmetic ^Ican > Geometric 'Mean Arithmetic Mean of ?h*'' powers > m^^ power of A. ... ... -.. Scales OF Notation ... Theorem concerning Sam or Difference of Digits
330 332
^l.
...
334^i2
...
Theory OF NuMRERS Highest Power
3i7 3^9 3053r)9
of a
Prime... ...
ji
contained in \m^... ......
...
... ...
Format's TheoremWilson's TheoremDivi.sors of a
... ......
.......
...
......
...
S,
Number (livisil.lebv2 + l
...
371 374
380
SECTION TIT.^THHOKY OF EQX'ATTON.S.... ... Factors OF AN EyuATiox To compute /(a) numcricallv ... Di.scriminati(m of Roots...
...
-..
'^'"^
4i3... ..
...
...
^O'J
Descartes' Rule OF SicNs
The
DEiiiVED Functions
of/ (.}) ... To remove an assigned term To transform an equation ......
... ......
..
-..
...
... ... ...
.....
410 424 428 4304.32^^''
Ei^TAL Roots OF an EquationPra-tical Rule
...
.
XIV
CONTEXTS.No.of
Article.
Limits OF THE Roots
448452 454 459 4GG 472 480
Newton's Method Rolle'fi Tlieorem
Newton's Method OF DivisorsRECiPROC-\Ti
...
...
...
...
...
Equations
Binomial EquationsSolution of .^"1
=...
bj Do Moivre's Tlicorcm... ...
Cubic Equ.viions
...
...
...
...
Cardan
s
Method...
Trigonometrical Method
...
...
...
...
483 484 489492 496 499 502
Biquadratic Equ.ations Descartes' SolutionFerrai'i's Solution
... ...
... ...
...
......
... ...
... ...
...
Euler's Solution
Commensurable RootsIncommensurable Roots
......
Sturm's Theorem Fourier's Theoi-em
... ...
......
......
... ... ......
... ... ......
506 518 525 5275285.30
Lagrange's Method of Approximation
...
Newton's Method
of
Approximation...
......
Fourier's Limitation to the same
...
Newton's Rule for the Limits of the Roots
Theorem Horner's MethodSylvester's
..
...
...
...
532
Symmetrical Functions of the Roots of an Equation ... ... ... ^ms of the Powei'S of the Roots ... ... Symmetrical Functions not Powers of the Roots... The Equation whose Roots are the Squares of the Differences of the Roots of a given Equation Sum of the m"' Powers of the Roots of a Quadratic
533
534 538
Root of an Equation through the Sums of the Powers of the Roots E-KPANsioN of an Implicit Function of a;to the
Equation Approximation
545 548551
Determinants
...
Definitions
55-i
General Tlieory
To
raise the
Order
of a
Dotonninant
Analysis of a Deterniinaut ... Synthesis of a Detcrminiuit
556 564 568 569 570 574 575 576
Product of two Determinants of the 7i"' Onlei Synimetrioal Determinants .. Reciprocal Determinants ... Partial and Ci>niplcnu'ntary DcierininnntH
CONTENTS.
XV\u.Aril...fI.'.
Theorem
of a Partial Ik-ciprocal Dclonuiiiunt
...
...
.')77
Product of DiU'ercuce.s of /i Quantitius ... ... ... Product of Squares of UiiTereiices of samo ... ... Rational Algebraic Fraction expressed as a UcleniiinantEli.mi.n.mio.n
578 5795sl
Solution of Linear Ecjuations Orthogonal Transformation
...
... ...
... ...
...
rs2r)Sl.r).s5
...
... ......
Theorem
of the
?i.'2"'
Power
of a Deteniiinant...... ......... ...
Bezout's Method of EliminationSylvester's Dialytic
5SG 587 5H85'J3
Method
... ...
...
^lethod by Symmetrical Function.^
...
......
Eliminatiox BY
llu;iii:sT
Cu.MMON Factou
...
SECTION IV. PLANE TRIGONOMETRY,Angular Measuremext... ... ... ......
...... ...
Trigonometrical Ratios ... ... ... ... ... Formula) involving one Anglo ... ... ... Formula; invoMng two Angles and Multii)lc Angles Formula-" involving three Angles ... ... ... Ratios OF 45, G0=, 15, 18, 1
Formulas forProof thatSina;TT
tliois
calculation of
......a;
......
792 795
iucomracnsural)lc
... ...... ... ...
=
?i
sin
(a;
+ a.). Series
for
...
......
790
SumSin
of sines or cosines of
Angles in A. P.
Exi^ansion of the sine and cosine in Factors
...
and cosnf expanded in Factors in Factors involving (^ Sin(? and cos e' 2cos6 + e"' expanded in Factors1/0
...
... ... ...... ...
.........
...... ...
De
Moivre's Property of the Circle......
Cotes's Properties
......
......
.........
...... ... ... ...
Additional Formulae
......
Properties of a Right-angled Triangle Properties of any Triangle.........
... ...
......
Area
of a Triangle
...
...
800 807 808 815 817 819 821 823 832 835 838841 850 859
Relations
between a Triangle and the Inscribed, Escribed, and Circumscribed Circles ... ... Other Relations between the Sides and Angles of a Triangleof the Solution of Triangles... ......
Examples
SECTION v.SPHERICAL TRIGONOMETRY.Introductort TheoremsDefinitions
870 871
Polar Triangle
Right-angled Triangles Napier's Rules
881
Oblique-angled Triangles. Formula) for cos a and cos A The (S Formula) for siniJ, sinJa,
=n("ACi-Mnillcl (o
n-ill-^7
Qh'=rrrj
"^^n^i^ll'-'l^
= n an.i (>v = :'.".'
.
Expansions of Lmplicit Functions
151G-7
Lagrange's, Laplace's, and Burmann's theorems, 1552, 1550-03Cayley's series for
--
...
...
...
15551-'"-
Abel's series for if>{x-\-a)
...
...
...
.
Indeterminate FormsJacodians......
1580... ......
...
1^*^^
Modulus of transformation
...
...
lOUt
XXVI
CONTENTS.No. of Article.
QiAViicsEuler's theorem..
1620......
...
...
1621
Eliruinant, Discriminant, Iuvariant,Covariant, Hes.sian
Theorems concerning discriminants Notation ^ = 6c-/, &cInvariants
...
...
...
Cogredients and EmanentsImplicit Functions
...
...
...
...
1626-30 1635-45 1642 1648-52 1653-517001725 1737 1760
One independent
variable
...
... ... ... ...
... ......
...
Two
independent variables
...
...... ...
... w independent variables Change of the Independent Variable
...
Linear transformation
...
......
......
......
...... ...
Orthogonal transformation
Contragredient and Contravariant
Notation z=p,&:,c.jq,r^s,t...
Maxima and Minima One independent
...
...
...
1794 1799 1813 181518301841 1852 18491862
variable
... ...
... ...
........
... ... ... ......
Two
independent variables
Three or more independent variables...Discriminating cubic... ... ... ...
... ...
Method
of
undetermined multipliers
Continuous maxima and minima
...
...
...
1866
SECTION IX. INTEGRAL CALCULUS.IntroductionMultiple Integrals
19001905
Methods of Integration
Parts, Division,
By
Substitution,
Rationalization,......
Partial fractions. Infinite sei'ies
...
1908-191921
Standard Integrals
...
...
Various Indefinite IntegralsCircular functions
...
...
...
...
...
...
...
... ......
...
Exponential and logarithmic functionsAlgebraic functions...
.........
1954 1998 2007 2110 2121-47 2127 2148 2180 2210 2213 2214
...
...
Integration by rationalization...
...
...
Integrals reducible to Elliptic integralsElliptic integrals
approximated to...a-,
... ...
... ... ......
... ... ...
Successive Integration
...
Hyperbolic Functions coshInverse relations
sinh-i;,...
tanh.r.........
...
...
Geometrical meaning of tanh
l
...
F(>i;.m.s...
24512571
Integration of Circular Logarithmic and Exponential Forms
Miscellaneous Theoremsformula)
Frullani's, Poisson'.'^, Abel's,
Kummer's, and Cauchy's2700-13......
Finite Variation of a ParameterFourier's formula...
......
...
27142726 2743 275727)
...
...
...
The Function \P{.c) Summation of4/ (.)
series
by the function
/'
(.t)
...
......
as a definite integral independent of \p{l)
Nu.merical Calculation of log r{x)
2771
Change of the Variables Multiple Integrals-
in a
Definite JMl'ltiple I.ntegral
2774
Expansions of Functions in Converging SeriksDerivatives of the nth order...... ... ... ...
...
... ......
2852 2911 2936 2955 2991 2992-7
Miscellaneous expansions
...
Legendre's function
X
...
...
Expansion of Functions
in
Trigonometrical Series
...
Approximate Integration ^lethods by Simp.son, Cotes, and Gauss
...
...
SKCTION X. CALCULUS OF VARIATIONS.Functions of one Independent VariableParticular cases...... ...... Other exceptional cases Functions of two Dependent Variables
......
......
......
... ... ...
3028 3033 30453051
... ...
Relative
maxima and minima
......
... ...
3069 3070
Geometrical applications
...
...
XXVlll
CONTENTS.No. ofArticle.
Functions of two Independent VariablesGeometrical applications
......
...
...
Appendix
...
...
...
3075 3078
Oil the general object of the Calculus of Variations...
Successive variation
...
...
...
...
...
Immediate integrability
...
...
...
...
3084 3087 3090
SECTION
XL DIFFERENTIAL... ...
EQUATIONS......
Generation of Differential EquationsDefinitions and Rules...
... ......
...
Singular Solutions
...
...
... ...
... ...
......
First Order Linear EquationsRiccati's Equation
Integrating factor for il/dc+iVfZi/......
=... ... ...
First Order Non-linear EquationsSolution by factors...... ... ...
......
......
3150 3158 3168 3184 3192 3214 3221 32223236 32373238 3251
Solution by difFei'entiation
...
Higher Order Linear EquationsLinear Equations with Constant Coefficients... ...
Higher Order Non-linear Equations
...
...
... ... ... Depression of Order by Unity... Exact Differential Equations ... ... ... ... Miscellaneous Methods ... Approximate solution of Differential Equations by
3262 3270 3276
Taylor's theoi'em
...
...
...
...
...
32893.301
Singular Solutions OF Higher Order EquationsEquations with more than two Variables......
...
Simultaneous Equations with one Independent Variable... Partial Differential EquationsLinearfirst
3320 3340
order P. D. Equationsfirst
...
...
......
Non-linear
order P. D. Equations
...
3380 3381 3399
Non-linear
first
order P. D. Equations with more...
than two independent variables Second Order P. D. Equations
...
...
Law
of Reciprocity
......
......
... ...
... ...
......
... ......
3409 3420 34463470 3604
Symbolic Methods
Solution OF Linear Differential Equations BY SeriesSolution by Definite IntegralsP. D. Equations with...
...
...
...
more than two Independent Variables ... Differential Resolvents of Algebraic Eqlaitons
3617 36293631
CONTENTS.
XXIXXo. of Article.
SECTION
XII.
CALCULUS
OF FINITE'^7(^6
DIFFERENCES.F0KMri,.K KOR FlKST AND uth UlKKKKKNCF.S
Expansion by factorials
.........
... ...
... ... ... ... ...
li/.^O
Gcnemting functions ... The operations 1/, A, andHerscbel's theorem...
...
3732373-5
(/.r
.........
...
3/.)7
A
theorem conjugate to Machiuriu's
Interpoi-ation
Lagrange's interpolation formula
...
...
...
3759 37G2 370H^^^72
MhXHANlCAL QUADRATURK Cotes's and Gauss's formula3 ... Laplace's formula
... ...
......
...
... ...
3777 o/lH'^781
Summation of Series Approximate Summation
3820
SECTION XIII. PLANE COORDINATE GEOMETRY.Systems of Coordinates
4001-28
Cartesian, Polar, Trilinear, Areal, Tangential, and
Intercept Coordinates
ANALYTICAL CONICS IN CARTESIAN COORDINATES.Lengths and AreasTransformation of Coordinates The Right Line Equations of two or more right
^032 40484(>
lines
General Methods Poles and Polars
4110 4114 4124 ^^^G 4U1'^^^^
The CircleCo-axal circles
The Parabola The Ellipse and HyperbolaRightline
and
ellipse
4250 4310433t.
Polar equations of the conic
Conjugate diameters
The
Determination of various angles Hyperrola referred to its Asymptotes
The rectangular hyperbola
4o4b 4375 4387 439-
.
XXX
CONTENTS.No.of
Article.
The General Equation The ellipse and hyperbolaInvariants of the conic
The parabola Method without transformation
of the axes
.
.
Rules for the analysis of the general equation Right line and conic with the general equationIntercept equation of a conic...
Similar Conics
4400 4402 4417 4430 4445 4464 4487 4498 45224527
Circle of Curvature
4550
Contact of Conics
CoNKOCAL Conics
ANALYTICAL CONICS IN TRILINEAR COORDINATES.The Right LineEquations of particularlines
4601
and coordinate
ratios
of particular points in the trigon
Anharmonic Ratio The complete quadrilateral The General Equation of a ConicDirector- Circle...
Particular Conics Conic circumscribing the trigonInscribed conic of the trigonInscribed circle of the trigon... ...
General equation of theNine-point circleTriplicate-ratio circle
circle...
...
Seven-point circle
Conic and Self-conjugate Triangle... On lines passing through imaginary pointsCarnot's, Pascal's, and Brianchon's
...
Theorems
4628 4648 4652 4656 4693 4697 4724 4739 4747 4751 4754 47546 4754e 4755 4761 4778-83
The Conic referredContact
to two Tangents and the
Chord of4803 4809 4822 4829 4830 4844 4870 4907
Related conics
...
Anharmonic Pencils of ConicsConstruction of Conics
Newton's method of generating a conicMaclaurin's method of generating a conic...
The Method or ReciprocalTangkntial Coordinates
Polars...
Abridged notation
CONTKNTS.
XXXINo. of Article.
On
Tin:
1m kksection
ok two
Conics......
Geonictricftl^mcanin^ of v/(- 1)
'I'-^l^'
The MethopTo
of Pkojection
'^'-l
Invariants anp Covakiantsfind the foci of the general conic.....
41K5G...
5008
THEORY OF PLANE CURVES.Tangent and Normal Radius of Curvature and Evolute Inverse Problem and Intrinsic Equation Asymptotes Asymptotic curves Singularities of Curves ... Concavity and Convexity-''l^'*^
-"il-^^
51G0 51G7r.l72
...
...
......
5174'
Points of inflexion, multiple points, &c.... ... Contact of Curves Envelopes Integrals of Curves and Areas Inverse Curves
......
5176-87 5188 5192 519G5212
...
...
...
...
...
...
Pedal Curves Roulettes
......
... ...
... ...
...
...
...
Area, length, and radius of cui'vature...
......... ...
...
5220 5229 5230-55239
The envelope
of a carried curve... ......
......
Instantaneous centreHolditch's theoremTrajectories
...
...
o243 5244 ^24G524752-*8
Curves of pursuitCausticsQuetelet's theorem
...
...
...
Transcendental and other Cuhves
...
...
5-49
The cycloid The companion
5250to the cycloid...... ... ... ... ... ...
......
525852G0 52G GG 5273 52795282
Prolate and curtate cycloids
Epitrochoids and hypotrochoidsEpicycloids and hypocycloids...
...
...
The The The The The The The
CatenaryTractrix
...
Syntractrix
Logarithmic Curve
Equiangular SpiralSpiral of Archimedes
5284 528852965302
Hyperbolic or Reciprocal Spiral
XXXnThe Involute of
CONTENTS.No. of Article.tlie
Circle
...
...
...
...
-5306
TheCissoid The Cassinian or Oval of Cassini
5309......
......
... ...
5313
The The The The The The The The
Lemniscate
...
...
6317
Conchoid
Lima9on
...
...
...
... ...
... ......
..
5320 53275335
Versiera (or Witch of Agnesi)Quadratrix.........
...
...
...
Cartesian Ovalsemi- cubical parabola...... ... ... ...... ... ... ...
5338 53415359
...... ...
folium of Descartes...
5360
Linkages AND LiNKwoRK
Kempe's
five-bar linkage.
Eight cases.........
Reversor, Multiplicator, and TranslatorPeaucellier's linkage......
... ...
...
The The The The The
six-bar invertor
... ...
eight-bar double invertor
......
Quadruplane or Versor Invertor
Pentograph or ProportionatorIsoklinostat or Angle-divider
...
...... ...
... ...
A A
linkage for drawing an Ellipse
5400 5401-5417 5407 ... 5410 ... 5419 ... 5420 ... 5422 ... 5423 ... 5425 ... 5426 ... 5427
linkage for drawing a Lima9on, and also a bicircular quartic...... ... ... ... ......
......
A linkageOn
for solving a cubic equation... ...
5429
three-bar motion in a plane...
......
The Mechanical Integrator The Plauimeter
5430 5450 5452
SECTION XIV. SOLID COORDINATE GEOMETRY.Systems of Coordinates The Right Line...
...
...
...
...
The PlaneTransformation of Cooi;niNATES... ... The Sphere The Radical Plane
... ...
...
......
...
5501 5507 5545 5574
...
...
...
5582
Poles of similitude
...
... ...
......
...
.........
Cymxdrical and Conical SurfacesCircular Sections...
... ... ...
5585 5587 5590 5596
...
...
Ellipsoid, HvriiRUOLOiD, and Paraboloid
...
...5590-5621
CONTENTS.N... of Article.
Centrai,
Qcapiuc Surfackdiainotnil plaiu's.........
Tangent and
......
... ...
Eccentric values of the coordinates... ... ... CoNKOCAL QuAi'urcs Reciprocal and Enveloping Cones Thk Genkral Equation of a Quadric
5026 5038''^^0
......
......
......
56G4 5073'''""!
Reciprocal Polars
Theory of Tortuous Curves The Helix General Theory of Surfaces
'"^^Jl
575i;
......
... General equation of a surface ... Tangent line and cone at a singular point
......
5780
5783^^795
The
Indicatrix Conic
Eulei-'s
and Meunier's theorems...
...
...-.
...
5806-95826 5835
Curvature of a surface...GeodesicsInvariants
...
...
Osculating plane of a line of curvatnri!...
...
...
...
-..
...
.
Integrals for Volumes and SurfacesGuldin's rules......
...
......
...
5837-48 ^856 5871587i)
...
...
.
Centre of Mas.s Moments and Products of Inertia Momcntal ellipsoid Momental ellipseIntegrals for
5884 5903 5925-405953 5978
moments
of inertia
...
...
...
Perimeters, Areas, Volumes, Centres of Mass, and Moments of Inertia of various Figures
Rectangular lamina and Right
Solid...
...
...
6015!^ri7 centimetres per second. In latitude X, at a liciglit h above the sea level, -OOOOO:]/;) centimetres per second. cos g = (98O-0U56 Seconds ])endaliim = (iJ9-85G2 -2536 cos 2\ -0000003 h) centimetres. THE 7';.17i"i7f. Semi-polar axis, 20,854890 feet* = G-3;.4ll x lO^centims. 3782t x 10" 20,9_'G202 * = Mean semi-equatorial diameter, 39-377780 x 10' inches* = TOOOlOO x lO" metres. Quadrant meridian,In the lutitudc of London,cj
2-.'')028
2/\
oi"
Volume, r08279 cubic centimetre-nines. JIass (with a density 5g) = Six gramme- twenty-sevens*
nearly.
These dimensions nro
liikcn frjiu C'larko'a
"Geodesy," 1880.
MATIIEMA TICAL TA BLES.
Velocity in orbit
=
2033000 ccntims per
sec.
Ohii.iuity, -2:f 27' lo".
Aiii^ular vclucily of rotation
13713. -01079. I':cCentricity, e Prounession of Apse, U"-2.'). Precession, t>0"'20.* Centrifugal Ibrce of rotation at tlio equator, ;>-3'.)12 dynes per ^nunnio. Force of sun's attraction, -0839. Force of attraction upon moon, -2701. 2H0. Katio of (/ to centrilu^ral force of rotation, g rw* '.'.* Aberrat i'
l"b2.*
Approximate meanTropical year, Sidereal year,
distance, 1>2,UUOUUO miles, or l"'i8 centimetre-tliirleens.t
3Go2422l6
days, or 31,550927 seconds.
305-250374 Anomalistic year, 305-259544 days.
31,558150
Sidereal day, 8010
!
seconds,
0-98 10^ granimes. Earth's ma.'^s X -011304 TJllJ M0UN.Uas8 Horizontal parallax. From 53' 50" to 01' 24".* month, 29d. 12h. l-lin. 2878. Sidereal revolution, 27d. 7h. 43m. 1 l-40s. Lunar Greatest distance from the earth, 251700 miles, or 4U5 centinieire-tens.
=
=
Least
Inclination of Orbit, 5 9'.
225000 303 Annual regression of Nodes, 19 20'. Hulk. {The yt'ar+l)-^19. The remainder is the Guhlen Number. The remainder is the Ejiact. {Tlie Uulih'ti Number 1) X 11-^30.
GRAVITATION. Attraction
x l-54o x It/ The mass which at unit distance (1cm.) attracts an eijual mass with unit force (1 dn.) is = v/(l-543x 10^; gm. = 3iV28 gm.j;-'
between masses m, m' at a distanco /
)
~
mm
clvues
Tr.rr^/i!. Density at 0C., unity
;
at 4,sq.
1
0000l3 (Kupffer).
Volume
elasticity at 15, 2-22
Compression Descamps).i is
for
1
X iV". megadyue per
cm.,
4-51x10-* (Amaury andmass of water from0 to
The heat requiredproportional to
v(n'('i't
(I
Scries info a dnitiuucil Fractiini....
series
i + ^ + :!l +second, and1
+with;/
('(jual;
to a continued fractionth(^ first,
1^ 0^'^'^),
-|- 1
com-
poneiits
//-t-l"' coinj)on('iitsu'i
being
ir,viii-{-i(.r,,
,.r-i.
Here X
=
=
7
satisfy ilie equaticjii'
;
,/
~ = 7+.r,
fiinii.Nli :ill
tilt'
solutions.
U//
*
The simultiinoous vahu'S
of
/,
und--21-
will he as follows
:
.,
,j
t=-l) = -5 = -rA
-i -2 -9
-:i1
-I7
1
-J
:;
107
l:{
101.-)
I'.'
-5-1and
3
11
19
positive integral solutions is infinite, ami the least positive integral values of x and ij are given by the limiting value of /, viz.,
The number of
t>-\thatis, t
t>-\-'
mast be 1,
0, 1, 2, 3,
or greater.
190 It" two values, a and /3, cannot readily be found inspection, as, for example, in tlie equation17.t'
by
+ 13// =
14900,the re iiiaiiiliKj frac-
dlridr
In/t,
f/ic
huisf roi'ffirient,intc/jer;
and equate
tions to
an
thus
*+'+ if ="+!;,4a 2Repeat the process;
'"
=
l-.it.
thus
4
4
Pat
68
ALGfEBBA.
Here the numberintegers^
of solutions in positive integers
is
equal to the
number
of
lymg between
X.
,
7
and
,
1137 --
;
or
~ Tq
^^^ ^^Tf
;
t^^t
is,
67.
191 Otherwise. Two values of x and y the following manner:
may be found17 y^. 33[By
in
Find the nearest converging fractionThisis
to
(160).
.
By
(1G4)
we have
17x3-13x4 =Multiply by 14900, and change the signs;17
-1.
(-44700)
+ 13a,
(59600)
=
14900
which shews that we may take ^and the general solution
,
( /5
= =
-4470059600^^^
may bex
written
=
-44700 + 13/,59600-17^.y8.
y=192The valuesof x and
This method has the disadvantage of producing high values of a and
//,
inc,
satisfy the
equation ax + bi/
=
positive integers, which form two Arithmetic Pro-
gressions, of which h and a are respectively the See examples (188) and (189). differences.
common
193
Abbreviation of the method in (169).:
Example
ll.i; 18;/
=
63.
Put X
= 92,
and divide by 9
;
then proceed as before.
194
To
ohld'ni iiifriinil s(thitioN.s'
nf
(H-\-f)t/-\-rz
=
(I.
Write the equation thusax -{-III/
= (J
cz.
Put successive integers for
;:,
and solve for
.r,
//
in encli cnse
ItEDUGTION OF A QtlADJiATfC srUD.
GO
TO
Iv'KDlTCF]
A QUA1>HATI(^ SLIHI) TO CONTINUED FRACTION.
A
195
EXAMI'I-K
:
^29=4
5+v/29-r,
=
'^
5-h^
,29 + 5'5
y29 + 5_ ^, v/29-:5^ '^^.
4
"^^ ,'29 3' +
~5
^.,
5
^^29 + 2',
v/29 +"
2_~
5
"^
v/29-3_5
4:>'
^
/29 4 + 3_ ^/29
g ^
+ "4,
v/29-529
_
^V^29 + 2,
^
+
v^29+.yv/29 +5'
+ 5 = 1U+
V
-5 =
10
+
Tlio (iiiotients 5, 2, 1, 1,2, 10 arc the gTcatest integers
contained
now
in the quantities in the first cohimu. recur, ami the surd \/29 is equivalent to
The quotientstlie
continued
fraction
5+511
2+1+1+10
1_ 1_ 1_
1_ 1_ J 2+ 10+ 2+ 1+ 1+1
1
]
2
+ c^c.be
The convcrgentsT'
to v/29,
formed as727135'
in (IGO), will
27
7013'
2'
3'
5'
1524 283'
2251 418'
3775 9801 701' 1820'
Note that the last quotient 10 is the greatest antl twice the first, that the >r tlian1,
208209The The(7o,
I'^or all
values of
rtr
is
.
'Hi'-
number
of (juotientsis2(i,,
lastfirstr.,
(luotient
cannot be greater than 2a'l and after that the terms repeat.is
complete quoti(>nt that
repeated
is
^
\
'\ and
7-0,
commence each
cycle of re}eated terms.
72
ALGEBRA.
210c,_2
just given.
220
//
+ + //- = ^r :^-^-.r-\-x.v = fr 'H/ + .*//-r^.^'
(1),(2),
(3);(4).
:5(//.r
+
.v.'
+ ,o/)-=r
:l/n--\-2r',r-\-2'rlr-a'-b'-c*
iMAniXAUY
i),
and subtract
tlie
square of
(1).
Result
X
(3./'//^
- Jf -if-
::'')
= h'(-2-l);Result1
D=4a;
i.
(a 1)'
zj.
+
27
3TTT H
(a;-l)
-l
,
H
rii" +2
237
Thirdly. When there
is
a quadratic factor of imaginary
roots not repeated.
Ex. Resolve ,ttw^2-.Here we must assume
,
ix
into partial fractions.
Ax-{-B(a5+l)(j!
Cx + Dx'
+ + l)a!
a;^
+l
+ x+l'
x-i-l and X- + X + 1 have no real factors, and are therefore retained as denominators. The requisite form of the numerators is seen by addingtoo'ether
two simple ^
fractions, such as
x+b
- ^
x+d
r~,-
Multij)l}iiig up,1
we have(Ax + B)a;-
the equation(x'
=
+ x + l) + {Cx + D)z.
(x'
+ l)
(1).
Let
+l =
0;
x^ = I.;
Substitute this value of1
x- in (1)
repeatedly
thus;
=
(Ax + B) X
or
Equate
coefficients to zero
;
= Ax' + Bx = -A + Bx Bx-A-l = 0. 5 = 0, ^ = -1..'.
Again,
let.-.
ar
+ + l=0; x-=-x-l..r;
Substitute this value of1
x^
repeatedly in (1)
thus
=
{Cx + D) i-x)
or
Equate
coefficients to zero
;
= -Cx'-Dx = (G-D)x + C-l=0. = 1, thus^'
Cx + C-Dx-
1)=XT ^^'"'^^(..^
1.
1
_
=
''+1
_.
^
+ l)(x^ + * + l)there
.tHx + 1
a-Hl
238
Fourthly.
When
is
a repeated quadratic factor
of imaginary roots.
Rv "Resolve
40.^'
103
^
i)artiiil
fractions.
;
80
ALGEBRA.Assume40.7;
-103
^
Ax + B(.r2_4x
_CxB _(.'?;--4a;
Ex + Fa;--'-4^
(x +
iy {x'-4x + Sy
+ 8y4-
+ 8)a;
+8
(.r+l)-
-^ + -^; +
l'{x
40.t;-103
=
{iAx + B) + {Cx + D)ix-ix + 8) + iEx + F)(x--4:X + 8y}
+ l)(1).
In the first place, to zero thus a;2=4a;-8. Substitute this value of x- repeatedly in (1), as in the previous example, until the first power of x alone remains. The resulting equation is;
+ {G + H(x + 1)} (x'-4x + Sy to determine A and B, equate rt;- 4a; + 8
40a;
-103=we
(17.4
+ 65) -48^ -75.a?
Equating
coefficients,
obtain two equations)
17^ + 65= 40 48^ + 75 = 103)'
f ^^--^^^^
..
,
A =
2l.
B=
Next, to determine and D, substitute these values of A and 5 in (1) the equation will then be divisible by a;^ 4a; + 8. Divide, and the resulting equation is
=a;'-
2x + l3+{Cx + B+(Ex + F)(x'-4x + 8)] (x +
iy(2).
+ {G + H(x + l)]{x'-4x + 8y
4a; + 8 again to zero, and proceed exactly as before, when Equate finding A and B. Next, to determine E and F, substitute the values of (7 and D, last found divide, and proceed as before. in equation (2) Lastly, G and are determined by equating a' + l to zero successively,;
H
as in
Example
2.
CONVERGENCY AND DIVERGENCY OF239Letbe a
SERIES.
(7, a+^ auy two convergency may be applied. Tlie series will converge, if, after any fixed term (i.) The terms decrease and are alternately })0sitive and
ai-{-a.^-\-a;i-\-&c.
scries,
and
consecutive terras.
The foUowino-
tests of
negative.(ii.)
Or
if
"(' n 1-1
is
always
(j
renter
than
some
(piantity
greater tlian unity.
SERIES.
81
(iii.)
Or
if
'''1
i.s
never
less tluui tlic corrcspoiidiii^ I'atio
+
1
ill
a
known(iv.)
coiivei\u:ing series.if
Or
l-^n)
is
always
tjreafrr
than some (juan-
tity greater(v.)
than unit3^if
[% is
tl'
and
iii.
Or
l^-^ii l]\og)iV^'j+i^
always
i/rrdfcr
tlian
some quantity greater than
unity.
240
The conditions(i.)
of divergency are obviously the converse
of rules
to (v.).
241
The
series
ai-^a.,x-\-a.iX^-{-&c.
converges,1
if
^^
always less than some quantity p, and x loss than
[By 239
(ii.)
242
To make
the,v
sum
of the last series less than an assigned,
(iiiantitv /sefficient.
make
less than
,
I'
hvincr the o^reatest co-
Grnrral Tltcnron.
243 If ('") be positive for all positive intec^ral values of .r, and continually diminish as increases, and if )n be any positive integer, then the two series/>
{m-)-\-m''(t>{m')-Y
arc either both coiivern-ent
diverofent.
244
Ajiplication of
tliis
theorem.
To
ascx,
be expanded in ascending powers of x in three different ways. First, by dividing the niiraerator by the denominator in tlie ordinary way, or by Synthetic Division, as shewn in (28). Secondly, l)v the metliod of Indeterminate Coefficients(2:32).
Thirdly, by Partial Fractions and the Binomial Theorem.
SERIES.
83
To expand byproceed as followsAssume'^''^ ,
tlie:
method
of Indeteriiiiii:ite CoefficiLiits
~\^'^'.
,
=
-1
+
^''-
+ C.>- + J).c' + E.v' + & c.nx''+GC'u;*-
4x-lUr = .1+
llx+
Cx--\-
Ex'+Gi*./;'
I2
- 2
+ + Pm
.
(in - m-
Scale of Relation
is
252The sum
1 -PI^V -JhO^ ... lhn^V''\
of n terms of the seriesfirst
is
equal to
253
[The
m
terms
l terms + the last term) m 2 terms + the last 2 terms) IhJC^ (first m 3 terms + the last 3 terms)piV(first tn
p^x^
(first
-~i>i-i'^'""^ (first
p,nX"' (the last
term + the last m \ terms) m terms)] -^ [lp^.vpocV^ ...
/>,cr"'].
If the series converges, and the sum to infinity quired, omit all " the last terms " from the formula.
254
is
re-
255
Example.
Requiredsum
term, and the apparent4'c
the Scale of Relation, the general to infinity, of the series
+ 14r + 40,v^ +six arbitrary
110,ii'^
+ 304^^-8o4/+sufficient to
...
Observe thatfor,
terms given are
determine a Scale;
of Relation of the
form lpx qx' rx^, involving three constants p, q, r, by (251), we can write three equatious to determine these constants The solution gives 110= 40p4- 14(2+ 4r\ namely,304 854
= llOp + 402 + 14r k = 304;j+110g + 40rJof Relation is1
p
=
G,
7
=
- 1 1,
r
=
6.
Hence the Scale
6. + ll.r 6.r^.found from (254), by putting40.i-''
The sumPi
= ^,
Pi
= 11
of the series without limit will be P3
=6,first
m = 3.th ree termsfirst
The-I-
6xthe1 l.r
two terms X the first term
= 4,c + 1 4.r + = 24^- 84a;* = + 44a;'4-10a:-
RE CURBING SERIES.
^^ 1
4.r-10x'G.i;
+ 1 Ix* -
+
!j>
wliere Ji, A.,, &g., are determined by i)uttiug successively in the equation1
=
1, 2, 3,
&c.
2 0^
+ 1)!
~(;?+2)!"^r(/>)!'^r(r-l)(i>-l)!^"'"r(r-l)...(r-7>+l)
277
"'
^
(>,
+ !)
^,'"
+ + +(^/
^/)"'
+ + 2^/)"'+... + +(^/ (^/
//""
B,(253).
283
i^,,^W//^--])^;/(-])0/-2)_^^^.^.^^^,^
Hv making 4^-=^
in (12r).
.etrrat*.
94
ALGEBRA.
284
The
series
-,-j,
(n-4)(n-5);5!
^~"T""^^ ,
_ {n-6){n-r,){n-7) ^*,
4!
(
^y.,
0i-r-l)(n-r-2)...(n-2r-[-l)
^^~consists of^^
or
terms, and the
sum
is
given by
/S'
= ==
n
if 71
be of the form 6m-\-S,be of the form6/?i
S
if
?i
+ l,
S>S^
if
?i
be of the form 6m,
n
=
n
if 7i
be of the form 6m_2.
Proof. By
(545), putting;)
= x^-y,q = xy,
and applying (546).
285
The
series
7i^-n (-!)'+
Hd^l^ {n-2yo
!
takes the values
0,
n\,
^n{n + l)\
according as rProof.
is
X
the result of substituting/3 =:=
for
ic
in
(291),
and making
=X
.
Tlien,
by
last, or
independently by induction,'Pj!]_
/(y + 1)
_
Ay)
1+1+1 +with
1_^ p^_ P2_...
j),
+ I+&C. = (y+m,r,
1) (y+/>]
294
In this result put y
= ^ and
-^
for
and we obtain by
Exp. Th. (150),
Or
of one series
the continued fraction may be formed by ordinary division by the other.
295
('"'
is
incommensurable,(17-1),
m and
n being integers.'
From
the last and
by putting x
=
INTEREST.If r11
be the Interest on 1 for 1 year,the inimber of years,
/'
the
I'l-incipal,n.
A
the auiouut in
years.
Then
296297
At Siiuple InterestAt
Compound
Interest
A = A =
P{l-^)n').
/*(t+r)".
%
(-^-i)-
.
1:
i\Ti:i:i:sr
AM)
.l\.\ ///'//vS.
99
298
But
if the payments of ") Interest be made 7 C times a year )
.
A
=
I'
h + -j^^^
If
A
be an amount due in-1.
11
years' time, and
/'
the ])resent
worth of
TheuInterest7*
299
At Simple
==
-j-^
.
By
(-200).
300301
At Compound
Interest
/'
.
,
By
(297).
Discount
= A- P.
ANNUITIES.302The amount of an Annuity of 1 in n years,at Simple Interest1
[
= = _
nA~
,,(,,_])>'
.^
^y
(82).
303
1
'resent value
of
same
,tA-hi{N-\)r'^yr
n^(09,j).
304
Amount
at
Compound \)
(
1+r)" '
^
,^-.
Interest
"(l+r)-l
Present worth of same
~ (l+r)-l'.
'' .
,
'b'
(-J^'*^*)-
305
Amount whenmentsof
the pay- ^
(
j
1
_!1
Y"'
_1
Interest/ are made q times -|)er C
_ ___7_/
1
iw
(2:is).
'*i1
V''
annum
J
~
'
\
(/
1
Present
value of
same
=
100
ALGEBRA.
306
Amount
wlicn the payof the Annuity times per are made
)f
meuts
m
_m
( 1'
-f >)"i
I
~
[
annumPresent value of same
J
\{l-{-r)'-l}
^
l-(l+ r )m{(l-fr)i-l}
307
Amount whenterestis
the
In-
paid q times
and the Annuity times per annum
m...
"^J
V\
qi'
Present vahie of same
m
(i+v)-
PROBABILITIES.If ^11 tlie ways in which an event can happen be m number, all being equally likely to occur, and if in n of these m ways the event would happen under certain restrictive
309
in
then the probability of the restricted event hapconditions pening is equal to n-^m. Thus, if the letters of the alphabet be chosen at random, any letter being equally likely to be taken, the probability of a vowel being selected is equal to -i^q. The number of unrestricted cases here is 26, and the number of restricted;
ones
5.
they
events are not equally probable, all the divided into grou{)s of ccpially probable cases. The probability of the restricted event happening in each group separately must be calculated, anel tlie siun of these probabilities will be the total })robability of the restricted event liappening at all.
310
Ifj
however,
m
may be
I'UOHAIIILiriHS.
lol
ExAMPLK.
Tlicro are three bags A, B, and G.Acontains 2 white and .'}
black balls.
BC
3
t
4
5
bapf is taken at random and a hull bability of the ball being white.
A
drawn from
it.
Required the pro-
Hero the probability of the bag A being chosen probability of a white ball being drawn l-
=
=
J,
and the
8ub.sc([nonfc
Therefore the [jrobability of a white ball being drawn from
.1
~ ~And
3
5
15-
Similarly the probability of a white ball being1'
drawn from
B
X 37
3
~
l'
7
the probability of a white ball being
drawn from G
-1 ~.j2_
3 ^ 9
J*
- i ~ 27'427
Therefore the total probability of a white ball being drawn1
^
401945'
15
7
If a
be the number of ways in
wliicli
and
J)
tlie
number
of
ways
in wliieli it
an event can liappen, can fail, then the
311
rrobabilitv of the event lia])penin2r
= =
r.
312
l'rol)al)ihty of
the event failing
Thus
Certainty
=
1.
If p, p' be the respective probabilities of two iudcpcndcnt events, then
313
rrol)al)ility)}
(^f
both liappening/yo//i
=
pp'.
314315316
of notof one
happening == ipp'.
))
happening and one faiHng
of l)o(h failing
=
(!/>)
(1 />').
-
102
ALGEBlLi.
and the probability
If the probability of an event happening in of its failing q, then
one
trial
be
j>,
317
Probability of the event happening r times in n trials
=318
C{n, r)2fff-\
Probability of the event failing r times in n trials
=319nof 0;trials
C
{n, r) ^j" "''(/''.
[By
induction.
Probability of the event happening at lea>-.>+ +^' />i
+
lies
between the -Teatest
uiul least of
the fractions i^,
^,
...
-^, the (leiiominators being
all
of
the same sign.PuoOF.
Let
k be the greatest of the fractions, and
if
any other; thenk be the least
ar V"0.
332or,
.+.+ ...+ > y,7^~^,;Arithmetic mean
> Geometric mean.
Proof. Substitute both for the greatest iind least factors their ArithRepeat tlie process metic mean. Tiie product is thus increased in value. indefinitely. The limiting value of the G. M. is the A. ]\I. of the quantities.
333excepting whenPlJOOK'"
q:^'m+is
>
{^l+!i)'\
a positive pi'opei' IVaet ion.
?,'"
=("t'')"'[(l+.r)'"
+ (l-..)"'},
diere
.
=
"
-'.
Kinploy Hin. Tli
a
+b
334
":'+""'+..+":
> ^'.+".+ +".. y\
excepting ^vhen
///
is
a positive proper fraction.I"
;
106
Ahdi'JiiUA.
The ArUhmetic mean of the m"' poivers is Otherwise. greater than the m"' power of the Arithmetic mean, excepting is a positive proper fraction. when
m
Pkoof. Similarleft side,
to (332).
Substitute for the greatest and least on the
employing (333).
336then
If
-''
and
m,
are positive, and x and(l
mx
less
than unity(125, 240)
+ ciO-'"> l-mx.
337by
K
,1^
taking' x small
m, and n are positive, and n greater than enough, we can make
ra
;
then,
For X maybe diminished is > (l-\-xy\ by last.
until
l^nx
is
> {lmx)'^, and
this
338If X
If ^
be positive,
log {l-\-x)1,
log (l+.r)
>
1)0 |)ositiv(>
quantities,is
V/'
>
('i+I'f"'.
SinulaHy
a''
!,'.'>
{^'
f"*' + +','
'
These and similar theorems may be proved hy takinf^ lou^arithms of each side, and employing the Expon. Th (loH), Sec.
SCALES OF NOTATION.
342oftlie
It"
iVbe a whole number^
,
and,^ is
1;M-ft,
p
is
now prime
to 7,
and
a
prime to;
it
folhiws,
that
neither greater nor less than
that
is,
it is (.Hjual
to
it.
Therefore, &c.
351II
If
'^^ is
divisible(lb
by
c,
and aa'
is
not
;
then
h
must
be.
Pi;nOl'. Let
T
,
=7;
=l>
c
c is
Hut
'(
is
prime toIf.
but one of the factors, that factor(:ir.l)
divisible
354'/;
and
Therefore, if a" if j) be a prime
is divi.sihle
by ^^a.
i>
cannot be jirime to
it
must divide
355}
If
"It.
is
prime to
h,
any power of
m
mV!
p
F//^;
For there areit
'
factors in
which p
will divides
'.,
which
and will divide a second time divisions are eejuivalent to dividing
so on.b}'
The successive
ExAMi'LK.factors'"
Thel,
hitrliesfc
power
3,
''('''
tegers, including
,
the number of in' in prime factors which are less than u and prime to it, is;
Proof.
The//",
number&c.
of intogcrs
piimo
to
N
contained
in
ri"
is
n"-
Similiirly in
/',
Take the
])r()duct of those.
TUEORT OF NUMBERS.Also
113
tlio miinhcr of intcfjfors less tliaii mikI ])i-imo to ^fxScc.) is the ])roduct of the coiTcspoiuliiig miiuhcrs for X, ^[, &c. separately.
(Xx
374is
The number(y
of divisors of(/'
=
+ l) (v 4-1)
+ !)
...
N, incliidiiif^ For it is equal
1
and
^V itself,
to the
number
of terms in the product
(l+./
+ ...+r7'')(l+/.-h...+^")(l+r+...+'")---
where
a root of '" 1,
but neither
a, /3,
nor 7
= 1.
7
x^-l,''-!'Proof as in (475).
478
If
n
= m^
anda(i
be a root of x""! x^>^-a
= 0, = 0,
7 then the roots of x''l
r.--|3=0;will
=
be the terms of the product
(l+ + a^+
...
+-^)(l+/3 + /3-^+...+r-^) X(l + 7 + 7^"+... +7""')-
479 480is
a^" + 1 may be treated as a reciprocal equation, and depressed in degree after the manner of (468).
=
The complete
solution of the equation.1
-
-1 =(757)
obtained by
De Moi\Te's Theorem.by the formula
The
71
different roots are given0?
in
n which r must have the successive values71,
=
cos
V
1
sill
0, 1, 2, 3, &c.,,
concluding with ^
if
n be even
;
and with
-~
if
//
be odd.
CUBIC EQ UA TIONS.1
7
2
481
Similarly the n roots of the ofiuatiuu.r"
+
1
=
are given by the formula
n r taking the successive values 0, 1, 2, 3, &c.,'
u
up to
~^' ,
if
n be even
;
and up
to
,
if
)i
be odd.
482
'I'he
number
of different values of the product
is7/
equal to the least are integers.
common
multiple of
m
and
n,
when m and
CUBIC EQUATIONS.483Tosolve the general cubic equationa;^
+ jj.r + qx -f = 0./
Remove
the term
j^at^
by the method of (429)..v'^-\-q,r-\-r
Let the trans-
formed equation be
=
0.
484
Cardan
s
mcfhoiL
Thex
complete theoretical solutionis
of this equation
by Cardan's method
as follows
:
Put
=
i/-\-:i
(i.)
yH,v^ + (3v.v + 7)(y + ,v)
+r =
0.
Put
Si/::
+q =
0;
.'.
^
= - 3^
Substitute this value of //, and solve the resulting quadratic The roots are equal to 1/ and .r* respectively ; and we //^. have, by (i.),in
485
r
{-iWf+j^r+i-^-vj+f;}'
128
THEOBY OF EQUATIONS.real root at least,lij'-
Tbe cubic must have one
(400).
Let i be one of the three values ofof the three values ofj
j
^9"[
"^
\/ TT
"*"
'^
^'
(
^^ " "
I
^
\/
X
"^
'
486
Let
1, n, a-
be the three cube roots of unity, so that
a=-l487Then, since
y^, Viu^ = my+1m+ii,
and
ci'
=-
1-
- L yZs.
[472
I,
the roots of the cubic will beu'in-\-n)i.
am-ta'-n,
Now,
if in
the expansion of
I
2
^V
4
^
273
by the Binomial Theorem, we put
then we shall haveor else
= the sum of the odd terms, and = the sum of the even terms and = ^ v; m= + m = + v/ 1, and n = y v 1fx
V
;
/u.
y,
/u
fji
;
according ^^ \/ 'T
'^
^
i^ ^^^^
^^ imaginaiy.
By
substituting these expressions for in and(i-)
n
in (487),
it
appears that
488
If
V"
+
^2/^,cj^
^ positive, the roots of the cubic will be
/i + >'%/ 3,
fi
yvo.
r-
(ii.)
If
-r
"^
97 ^^ negative, the roots2/1,
will be
fx0,
+ y^S,:?/!,
vn/S./J
,2(iii.)
3
If
4since
+
t^ 27
=fi.
the roots are2ot,
-m;
m
is
now
equal to
489
'/^/'^'
Trigonometrical method..1'^
The equation
+
r/.r
+r =
may bewhen -p4
solved in
tlie
following manner, by Trigonometry,
+ 77= 27.>.r"--+
...
+/>
=
0, (l-Gl)
liaving
= 1
iiii^l
tlio
remaining coefHcients integers.
503and
tlie inte')> and the last divisor by the last remainder, changing the sign of each remainder before dividing by it, until a remainder independent of x is obtained, or else a remainder which cannot change its sign; then /(a^), /'('^
OF
nil': /vmi7;a'.s'
of tuf
norrrs.
m
is less tliau n,
the degree
oiJ'{.i).
Obtained by expanding by division each term in the vahio of/'(.i) given at (432), arranging tlie whole in powers of .r, and equating coeiricieiits in llie result and in the value ofy^^r), found by differentiation as in (1-21).
535
If "i
^0 greater than
//,
tlie
forimihi will be
Obtained by multii)lyinga, h,c,
/(./)
=
by
.'""",
substituting for
.i;
the roots
&c. in succession, and adding the results. these formula?.{.,)
= a,-\-a,x-\-n,x- + &cterms
(i.):
then the
sum
of the selected
^-illbe
.v=
;a"-"'t
a.i "
+ 6x' + c.r + = px + qx + r =0tZ
I
I
The
equations and their eliminantqx\-
arc?
r
'px^-\-qx--'rrx
= =and
2^
q
px^^-qx'^
= aa?-\-b3?-\-cx + d=+ ra?
pa
q r
a
b c
d
oa;*
+ 6' + car+c/.c
=0
b c
d
156
THEORY OF EQUATIONS.
588
III
Method of elimination by Symmetrical Functions.
coefficients of their first
Divide the two equations in (586) respectively by the terms, thus reducing them to the
forms
/Gr) =.r- + /)i^^-^+
...
-\-p^=
0,
(cv)
=(p
.v''
+
')
=
633634:
cot(.l
+ /i) =
MI'I/ni'LE AXOLES.
163
635
,
164
PLANE TRIGONOMETRY.
654655656657X
^
^
1
/A-o
A\
tan A 1 tau^
[631, 632
658
= 3sm A4< sinM, cos 3^ = 4 cosM 3 cos ^ 3tan J tanM tan 3^ = l-3tanMsiu
3A
By
putting
B = 24
in (627), (629),
and (G31).
659
sin
660
= sin" 4 sin" B = co^-B cosM. cos (^ + IJ) cos (^ - i^) = cos^ ^ - siu^ B = cos"-B sin" A.(A + B)sin
{A B)
From
(627), &c.
661
sin
662
sin
= v/l + Y+ cos = \/l c-ives tlie
ratios ol
168
PLANE TRIGONOMETRY.Proofs. sin 15sin 18''
696697
is
obtained from sin (45-30), expanded by (628).
from the equation sin 2x3.7;
cos
3.i-,
where x
698699known
sin 54^ from sin
=
3 sin
a;
-4 sin^r,
where x
= 18. = 18.
the ratios of various angles may be obtained by taking the sum, difference, or some multiple of the angles in the table, and making use of
And
formulae.
Thus12^
= 30-18,
7^
= ^,
&c., &c.
PROPERTIES OF THE TRIANGLE.
700701
c
= aQOsB-{-bcosA.ahc
sin'
A
sin
B
sin
Ca?
702
= 6'+c' 26c cos A.11.
Proof. By Euc.
12 and 13,
= b' + c'-2c.AD.26c
703
cos
A ^
=denote the area
If g
= ^H-6 +
g^
^^^
ABC,
704
.u4=^^I^^i-^, ^o4 = ^^^.[641,042, 703.0, 10,1.
705 706707sin
tan
A_J 7-VA.V
I{k'
L\ (M r {s-b){s-c).v(.y_)
A
=^Vbe
(.y a)
(sb)
{s
c).
[635, 704
A=
^ sin A = Vs (s-a) (s-b) (.v-c),=1
[707, 706
708
\^'2b-i'''-^2(-a--^'2irb--(i'-b'-r\
.
rifOl'EUTlh'S
nr riilASULKS.
iC'.t
Thr Trianii/r tun ILetr
('irrlt
= radius of inscribed circle,radius of escril)ed circle
r^=
touchingcircle
tlie
side a,
B = radius of circuinscribiu'709l
b roiii h
i^.,
A
=.
^
-1-
^.
+
-^
ncos
c
710
r =
[By
a
= r cot It + r cot-
-
.
711
>
712
A
170
PLANE TRIGONOMETRY.
SOLUTION OF TRIANGLES.Right-angled triangles are solved
by the formula?
718
e^=rt2+6-^;
la^e siu A719h,
,
=z=:
ecos^,h tau
\a
A
,
&c.
Scalene Triangles,
720
Case
I.
The equationah
sinwill
A
siu
B
[701
determine any one of the four quantities A, B, a, h Avhen the remaining three are known.
721 When,sides
The Ambiguous Case.in
I., two and an acute angle opposite to one of them are given, we have, from
Case
the figure,.
sin
C
^=
e sin
A.
'C
Then C and 180Alsoh
-C arccvo
the values of
C and C, by
(622).
because
= =
v'a-
c-
siu-
A
,
A 1) + DC.to
722
Wlicn an angle
1> is
be determined from the equation
....sill
/,
.
.
/;
=:
sin
.1,
a
and\o
''
is
u small fracLiousin
;
tlici'or
fiirular.1,
measure
of
B may
be appi-oximatcd
a by putting
(U^C)
sin
and using theorem
(rOC)).
.
,
SOL UTIOX OF TU I A M
i
L ES.
1
7
1
723./
('\^i'
TF.
AVluMi two sidesside ahis
/>,
r
and thebytlic
inclndcMl aiigl(!
are known,
tlie tliird
priven
formula[702
(r=when logarithms
+ (--'2hrvosA,
are not used.tlie
Otlierwise, eni])loy
followin
I.
726$
log{bc) and
+ c) may
and c are known, the trouble of taking out be avoided by employing the subsidiary angle
= tan"',c
and the formulatan
727Or
X(B-C)
=
tan (
,
^
)
cot
^
[C55
else the subsidiary angle
= cos"'
'''
and the formula[04:3
728If a
tan i
(B- G) =
tan' ^ cot ^J
be ri(|uirod without ealcuhit iug the aiiglis
/>'
mid
/',
we may
use the9(i0,
formula(^.
1
^^^- = ^^ tan = cot B^C^b + c,__2b-
A
743
JD=^cos^
.
isriisn>iAuy angles.
173
If
AD
III-
]H>i{H'n.licular to
BCsin
744745
AD
hi'
A//''
li'
sin
G + c'tantan
sin
7?
r'^
J}-tnnC7) -h
tan
(/
REGULAR rOLYGUN ANDRadius of circumscribing Radius of inscribed circleSide of polyo:oncircle
CIRCLE.
Number 746
of sides
= R. = = a. = n.r.,
n
=^2of
cosec
n
r
a = cot TT
Area
Polygon\na- cot
748
=
=
ItilV
sill
^^
nr- tan
USE OF SUBSIDIARY ANGLES.749To adapt a_hto logarithmic computation./;
Take
=a-h
tan"'
thenh
a^h =til
a sec'
6.
750
i^'or
take ^
= tan"h
us
a
av/2 cos (0 + 45)COS0
751
To adapt
a cos;
Ch sin Csin
to logarithmic computation.
Take
=
tan"' ^b
then
a cos
C6
= v/(a^ + i-) sin (9 C).of
[By 617
For similar instauces of the use
a subsidiary angle, see (72G) to (730).
752angle.If
To solve ax-22)X + i2
quadratic equation by employing a subsidiarythe equation,[
=
l)e
Hy lo
.
174
PLANE TRIGONOMETRY.
Case
I.
If a be
< ^r,x
put
P
^='^.,
sin''
B
;
then2^5
=
2pcos'^. and
siV
f-.
[639,640
Case
II.
If q beX
>", put
= sec-0;0),
then
=px
(Izki tan
imaginary roots.tan*;
[614
Case
III.
If q
be negative, put
^=and
then[644, 645
= Vq cot
2
y^ tan 2
LIMITS OF RATIOS.
753when
-g-=-r =9 vanislies.
'
For ultimately
^=4i = AP APe
l-
[601,606 q
754755
n sin
.
= ^ when n 1
is infinite,
gy
putting -- for
in last.
(co^~)
when]
n
is infinite..
Proof.
Put
(
l-siu"
^,
and expand the logarithm by (156).
DE MOIVRE'S THEOREM.756wherei
(t'os
a+/
sin a) cos /8+/ sinyS)...)
...
&c.).
= cos (a+)8+7+ = V 1. ByInduction, or
+
^iu
(a+^+7+sin>i^.
[Proved by Induction.
757Proof.
(cos 6-\-i sin ^)"
=
cosa,
n6+}/5,
by puttingtj-c,
&c. each
=
^ in (756).
Expansion of cosnO,
in iwwcrs sinO
and cosB.
758759parts.
c'osM^
==
cos"^-C^(/J, 2) cos"-'^^ sin-^
4-C(,4)cos"-^^sin^^-ctc.sin
n0
n cos""' ^ sin 6C{n,
Ji)
cos"'^ sin*^+&c.
I'liOOF.
Expand
(757) by Bin. Th., and oHang-(-(,:i)lM.r-g+Ac.all
In series (758, 7.59), stop ut, aiul cwiliulc, n. Note, n is here an integer.
fmns
willi indices grciiter
Letc^c.
s^
= sum(a
of the G{n, r) products of tana, tau/3, tany,
to n terms.sill...
+ ^+y+c^'C.) = cosa cosyS 762 i'Os(a-{-fi+y-\-Scc.) = cosa cos/S real and imaginary Pkooi'. By761e(|u:itinf;
(.v,-.v,
+ .v,-.^c.).
...
(1
-6',+.s-.U.).
jiarts in (7o6j.
Exjmnsions of the sine and cosine in powers of the angle
764
sill^=^--^+|:^-&C.
(.OS^=l-^ + -[^+el'C.=x employing,
Proof.
Pute'ir)
and (755).
766 768
= eos^+f siii^.=2 cos^.
e''
c''-\-r-'"
= cos 6i sin 0. t'''-e-" = 2/ sin ^.l
By
(150)
770
itaii^-^Expansion2"ofro.s-"
^
+ 'tau(9 _6
,^
av^(Z
sin,''
in cosines or sines of0.l9
midtiplcs of
772
'
cos"^
=
cos n9-\-n cos(//-2;('(//,;{)
+ C'(/',2)773^\'lleu//
cos (;/-!) ^4-
cos
(/
W lion
e is
r"')],
and
806(7Gt"'),
fsina
+ r,sin(+/5)
+c-sln(a
\-
n, I)
= }-.{,'' F {e'')-c-'' Fie-'')]their exponential values
Proved bv substitutingAc.
for the sines
and cosines
.
180
PLANE TRIGONOMETRY.Expansion ofthe sine2ci?" ?/"
and
cosine in factors.
807
'f
^"
cos n ^
+
y-""
=to
1^-- 2.r//cos^+/] \x'--2.vijQ0^{e^^-Vf
n factors, adding
o
to the angle successively.j^
Proof. By solving theqaadraticon the left, wegefcic=i/(cos?i^ + isinn0)". The n values of .r are found by (757) and (626), and thence the factors. For the factors of a'"y" see (480).
808
sin
?i.l
alone be
>
^.
pro-
duce
to
meet BC.is
The
sup})leineutary formula, by (871),'
>in
r
)/>
sill
r*
tan-=\/'I
A
/sill
A-
^
sill
Sill ys
A sill^
(5 r) -La)
whevQT
fi
1/ = i{a-{-h-\-c).I
/
I
\
Proof.
sin^ = \ (1 cos^l).
Substitute for cos
/I
from (872), and(673).
throw the nnmeratorfor cos -.
of the whole expression into factors
by
Similarly
or
by the
The supplementary formulae are obtained rule in (871). They are2
in a similar
way,
887ooo 888ooft 889
cos4 = JnA ri-T.i\ and transform by (()70-d72).,.
i
,
a
sill
190
SrUERICAL TRiaOKOMETR Y.GAUSS'S FORMULAE.
897(1)
smi(.4
+ 7J) _
(io^\{a-h)C0S-2C
cos ^6^
(2)
cos^Ccosi(^ + i^)sin^C
sin^c
(3)
_ cosi(rt+&)cosl^csin ^c
cosi(^-/J) _sini((< + ^)sill
^
C
From any of these formulae the others may be obtained by the following rule:
the sign of the letter B {large or small) on one side of the equaJion, and ivrite sin for cos and cos for sin
Rule. Change,side.
on the otherProof.substitute the
Takes
sin-^-
(^ + 7?)(88-i,
=
sin-^x4 cos }jB
+ coslA
sin iZ>,
values by
885), and reduce.
SPHERICAL TRIANGLE AND CIRCLE.Let r be the radius of the inscribed circle of ABG ; r the radius of the escribed circle touching the side a, and B, Ba the radii of the circumscribed
898
circles
;
then
(1)
tan r
= tan ^A sin (v ) =sni a.
^
(3)
SI
n^
sin?r^l siiioTi siiioC
(4)
2 cos
^A
cos
^B
cos I ^
cusS+cosPkook.
(^^
yl) + cos (S B)-\-liic.valueis
Tlio
first
found from
tlie
ri^lit-auglcd triangle OAF, in which The otliei- vahies I)y (881-892).
AF = s a.
spjiEuicAL
'nnASiii.i-: asi> ciikli:.
l'.l
899(3)
(1)
i;ni r
=
tail
,\/l
sin.v
=111 (.V
)
=T
i.s
\A sini/i
sin ^(7
oV_(.os.S-c()s(N-.l) + c'-^-')Proof. From tlie right-angled triangle O'AF', in which AF'= s. NoiK. The first two values of tan r may be obtained from those tan r by interchanging s and sa.of
900
(1)
tiiii
= _
tan
Tift
cos S
c()s(.S-.l)sill \7^
in
which z
O'/;/^
=
tt-.S.
192
SPHERICAL TRiaONOMETBY.
SPHERICAL AREAS.902wliereProof.
area of
ABC = (A-^B+C-tt) r- = Er E = ^+7i + C the spherical excess.tt,
By adding the three lunesABDG,
BGEA,
GAFB,
and observing thatget
ABF =IT I
CDE,
(
A+]l+^]TTIT
27rr'
=
27rr
+ 2ABC.
903
AREA OF SPHERICAL POLYGON,of sides,
n being the number
Area
= = =
{interior Angles
( 2)r^
tt]
r'
{277 Exterior Angles}
{27r sides of Polar Diagram} r.
The
last value holds for a curvilinear area in the limit.
Proof. By joining the vertices with an interior point, and adding the areas of the spherical triangles so formed.
904
GagnolVs Theorem.sin1 L^ c> tj
_
\/ {si" ^ ^^^ { s -T 2 cos
u)\
^\n(sr-j
h)\
sin(.y
r)y
-^a
cos ^b cos ^c
Proof. -Expand sin \_\{A + B)~},{Tr-G)'] by (628), and transform by Gauss's equations (897 i., iii.) and (669, 890).
905
LlhulUier's Theorem.
UuiE = y [tan k
tan
i (.s ^0
ian
I {.s^-h)
tan
i
{s-c)}.
Multiply numerator and denominator of the left side by Proof. {A+B) 2 cos 1 (A + B-G + n) and reduce by (6G7, 668), then eliminate i by Gau8.s'.s foniiulio (S!i7 i., iii.) Tnuisfonii by (()72, 673), and substitute
from (886).
'
roLYIIHIili-ONS.
\\y.\
Vi)\M\VA)\{()NH.Lettliciiuiiilx'i-
of fiiccs, solid
angles, and cdircs, of any
|)(.lylicdr tliroc lines drawn from the ann^les of a tii;in/>
and DC, describe a
circle.
It will
Suppose7>.
r
to
bo a point on the rcMpiirod locus.l'\
Join
7' witli
A,
J?,
C,
and
Describe a circle about PBC cutting Al' in cutting P77 in (1, and join AH and UF. Tlien
and anotlicr about AliF
TK- = AF'-A
A'-
=
=2>:
J7^--7U AC Ai'.vF (II. '1).
(by constr.)
= A V'-TA(HI.:3G).
.
AF
(III. Z^\)
=:
ur.rjj
Therefore, by h^'pothe.sis,f/
=
(VP
.
rn
:
PP'^
therefore
Z Z)7V?
= PGA
(VL
= GP rn = A D 2) = PZ''7>' (III.
:
T)B (by constr.)22)
;
=
PC//
(III. 21).
Therefore the trianglesportional to
DPP, BGP
DB
and DC.Ifl'>
Hence thetlie
are simihir; therefore construction.
DF
is
a
mean
pro-
964
CoE.
=q
bisector of
BC,
as
is otlicr\Ndse
locus becomes the pcrpciulicular shown in (1003).
965to
To
find the locus of a point P, the tangentscircles shall
from wliich10;3(3.)
two givenLet A,:
have a given
ratio.
(See also
be the centres, a, h the radii q the given ratio. Take c, so that c h p q, and describe a circle with va' c'. Find the centre -1 and radius ^l.V locus of P by the last proposition, so that the tangent from P to this circle may have the given ratio to FB. It will be the required locus.(rt
B
> fe), and p
:
=
:
=
Proof.
By hypothesis and constructionFTh-
q'
FT' + U'
ir'-o^ + BF'
,^-
_ AF'-AX'BP"
Cor.
Hence the point canto
which the tangents
two
circles shall
be found on any curve from have a given ratio.
966to
To
find the locus of the pointcircles are ecpial.
from which the tangents
two given
Since, in (965). wo have simplifies to the following:
now p
= q,
and
therefoi-e c
= h,:
the construction
Take
AN=
y(a--6'-), find in ^IP take
AB AN:
AC.
The perpen-
But, if the circles inter.sect, dicular bisector of PC is the required locus. then their common chord is at once the line required. See Radical Axis(985).
208
ELEMENTARY GEOMETRY.and Concurrent systems and
CoU'niear
nfj)oints
lines.
967 Definitions. Points lying in the same straight line are Straight lines passing through the same point are colUnear. concurrent^ and the point is called the focus of the pencil oflines.
Theorem.
If the sides of the triangle
ABG,
or the sides
produced, be cut bj any straight line in the points a, 6, o respectively, the line is called a transversal, and the segments of the sides are connected by the equation
968Conversely,collinear.Proof.
{Ahif
:
hC) {Ca
:
aB) (Be
:
cA)
=
1.
this relation holds, the points a, h, c will
be
Througli= AD:
any vertex
Ato
draw
ADAD
parallel to the opposite side transversal in D, then
BG,:
meet the
Ab
:
bG
Ca and
Be,
cA
=
aB
:
(VI. 4), which proves the theorem. Note. In the formula the segments of the sides are estimated positive, independently of direction, the sequence of the letters being prepoint may be supposed to travel served the better to assist the memory. from A over the segments Ab, bC, &c. continuously, until it reaches A again.
A
969(sin
By
the aid of (701) the above relationsin
may be put:
in
the form
ABb
:
bBC) (sin C^a
:
sin
aAB) (sin BCc
sin
cCA) = l
970if
If be any focus in the plane of the triangle ABC, and AG, BO, CO meet the sides in a,b,c; then, as before, {Ab bC) {Ca aB) {Be :cA) = l.: :
Conversely, be concurrent.Proof.versal
if
this relation holds, the lines
Aa, Bh, Cc will
By the trans::
Bb to the triangle AaG, we have (9G8) {Ab bG) {CB Ba):
x{aO
0.1)
= 1.
And, by the transversal Cc to the triangle AaB,
(Bc:cA)(AO: Oa) x(aG: CB)together.
= \.
Multiply these equations
;; ;
COLLINEAR AND COX('rnili:XT
SYSTIJ^fS.
200
Tf J>r, ca, ah, in tlic last figure, be produced to meet the sides of .l//Oin 1\ Q, R, then eac^h of the nine lines in the tiu-nro will be divided liannoiiically, and tlie points J\ (,',
971
R
^vill
be collinear.
Proof.
(i.)
Take hP a transversal
to
ABC;::
therefore,
by (^08),
{Crtlu-rcforo,(ii.)
:
PB) (Be:
:
cyl)
(Ab
hC)
==
1
by
(1170),
CP PB = Ca(AB:
uB.
Take
CP
a transversal to Abe, therefore
Be) (cP
:
Pb) (bCpb) (bC
:
CA)
1.
But, by (070), taking
for focus to Abe,:
(ABtherefore(iii.)
Be) (ep
:
:
CA)
=
1
cPTake
:
Pb
= ep
:
pb.
PC a
transversal to AOe, and b a focus to(.la:
AOc\
therefore,
by
(0G8
&
070),
aO) (OG
:
Cc) (cBCc) (eB
:
BA)I?-l)
andtherefore
(Apall tlio lines
:
pO) (OC
:
:
= =
1,1
Aa
:
aO
=
ylj)
:
i'O.
Thus
are divided harmonically.:
AQ QC the harmonic In the equation of (070) put Ab bC ratio, and similarly for each ratio, and the result proves that P, Q, R aro(iv.):
=
collinear,
by (008).
Cor.
If in the
lines will passProof.
same figure qr, rj), pq be joined, the three through P, (^, li respectively.of be to
Take
harmonic division
as a focus to the triangle abc, and employ (070) and the show that the transversal rq cuts be in P.
972of
If
any polygon
a transversal intersects the sides AB, lUl, CD, &c. in the points a, h, c, &c. in order, tlien
{Aa
:
aB) {Bb
:
bC) {Cc
;
cD) {Dd
:
(IE)
...
kc =
1.
Pkoof. Divide the polygon into triangles by lines drawn from one of the angles, and, applying (908) to each triangle, combine the results.
Let any transversal cut the sides of a triangle and three intersectors AO, HO, CO (see figure of V70) in tbe points A', B', C, a, h' c respectively; then, as before,
973
tlieir
,
,
(J7/Phoof. Each Take the versal.
:
IjC)
{Ca
:
a
B)
{/)','
:
c'A')
=
1.
forms a triangle with its intorsector and the transfour remaining linos in smvissioi for transversals to each trianMe, applying CJOS) symmctricallv, and ciMidiinr the twelve equations. 2 E.'^ide
210
ELEMEXTAllY GEOMETRY
974 If the lines joining corresponding vertices of two triangles ABC, abc are concurrent, the points of intersection of the pairs of corresponding sides are collinear, and conversely.Let tho concurront lines Proof. Take be, Aa, Bh, Cc meet in 0.transversals respectively to ca, ah the triangles OBG, OCA, OAB, applying (9G8), and tlio product of the three equations shows that P, E, Q lie on a transversal to ABC.
p
follows that, if the lines joining each pair of corresponding vertices of any two rectilineal figures are concurrent, the pairs of corresponding sides intersect in points which are collinear. The figures in this case are said to be in pprspectlce^ or in, homology, with each other. The point of concuiTcnce and the hne of collinearity are called respectively the centre and axis of perspective or homology. See (1083).it
975
Hence
976 Theorem. When three perpendiculars to the sides of a triangle ABC, intersecting them in the points a, b, c respectively, are concurrent, the following relation is satisfied ; and converse^, if the relation be satisfied, the perpendiculars areconcurrent.
Afr-bC'+C(r-aB--\-Bc'-cA''
=
0.
of this theorem, the concurrence of the three perpendiculars is readily established in the following cases: (1) When the perpendiculars bisect the sides of the triangle (By employing I. 47.) (2) When they pass through the vertices. (3) The three i-adiioflhe esci-ibed circles of a triangle at the points of So also arc; the radius of the contact between the vertices are concui-rent. inscribed circle at the point of contact wilh one side, and the radii of tho two
If the perpendiculars &c. 47). Examples. By the applicationProof.(I.
meet
in
0, then Ab''
bC-=
AO'-OC-,
escribed circles of the remaining sides at tho points of contact beyond the included angle. In these cases employ the values of the segments criven in (953). (4) The pei'j>endiculars equidistant from the vertices with three concurrent perpendiculars are also concun-cnt. (5) When the three perpendiculars from the vertices of one triangle upon the sides of the other are concurrent, then tho perpendiculars from the vertices of the second triangle upon the sides of the tirst are also concurrent.
Proof.tiou witlj
If A, B, G(I.
and A\
If,
C
angles, join
AB\ AC\ BC, BA\ CA, CR, and47).
are corresponding vertices of the triapply the theorem in conjuuc-
TRiAXiiLES rinrvMscjiunxt}
a rniAXi;rr:.
211
Trianii'lr.s'
of rnnstant species ptrctimsrribrd
to
a
trian'
any
ciri-lo
tliroii^Mi
and/''.
/',
and another
throii},diIj
r' iind //, intersectiiif^ the former in IJ and will cut the central axis in the required \>o\ut
Tlieir coniniou
chord
F
f.
Proof. IC. ID = IE. 1F= IC. 11/ from I to the circles are equal.
(111. 'M)
;
therefore the tangents
986
T}ii- cos pJ a and /3; then, by (993), XX'=YY', or a cos a that is, The cosines of the angles of intersection are inversclj/ as the radii of the fixed circles.
=
997in
The
and two together, intersect at a pointPimOF.
radical axes of three circles (Fig. 1046), taken two called their radical centre.(7
Letyl, B,
b/
t'lc
centres, a,
h, c
which the radical axes cut
JJC,
CA, AB.
tion ('J84) for each pair of circles.
Add
the radii, and X, Y, Z the points tlie equation of the definithe results, and apply {iUO).
Wrire
circle whose centre is the radical centre of three other circles intersects them in angles whose cosines are inversely as their radii (996).
998
A
i.\\i:iiSit>.\'.
Henco., thogonally,
ifit
this fourtli circle cuts
one of the others or-
cuts
them
all
orthogonally.a, ft,
999
'I'he circle
whicli intersects at angles
whose centres are .1, li, C and radii a, centre at distances from the radical axes uf thecircles,
y three fixed A, r, has itsiixed circles
proportional to/>
cos
ft
C BO
cos y'
C
(( COS a cos y CJi
(/'
COS
a
h AB
COS
/3
And therefore the locus of its centre will be a straight line passing through the radical centre and inchned to the three radical axes at angles whose sines are projiortional to thesefractions.Proof.for each
The result
is
obtained immediately by writing out equation
('J89)
p-.iir
of fixed circles.
1000 Definitions. diameter of a fixedwhose centreA-,
The Method of Inversion. Any two points F, V
situated on a
circle
and radius is so that 01\0r'= /r, are called immerse itoints with respect to the circle, and either point is said to be the inverseof the other.its
The
circle
and
centre are called the circle
and centre of
incersion, and. k the constant of inversion.
1001 If every point of a plane figure be inverted Avith respect to a circle, or every point of a figure in space witii respect to a sphere, the resulting figure is called the inverse or image of the original one. Since OB k 0B\ therefore: :
1002
OP
:
OP'
=
OP'
:
fr
= A^
:
0P'\
1003 Let D, jy, in the same figure, be a pair of inverse In the perpendicular bisector of points on the diameter 00'. VD\ take any point Q as the centre of a circle passing through 1), I)\ cutting the circle of inversion in R, and any straight Then, by (III. 3r.), line through in the points P, B. OB on OR- (1 000). Hence.
OB =
.
OU =
216
ELEMENTARY GEOMETRY.
1004two
(i-)
i^i
^
'"ii'e
pairs of inverse points lie(ii-)
inverse points; and, conversely, any on a circle.
circle cuts orthogonally the circle of inverand, conversely, every circle cutting anotlier orthogonally intersects each of its diameters in a pair of inverse points.
1005
The;
sion (III. 87)
1006
(iii-)
The
line
IQ
is
from which
to a given circle
the locus of a point the tangent is equal to its distance fi^om a
given point D.
1007 Def. The line IQ, is called the axis of reflexion for the two inverse points D, D', because there is another circle of inversion, the reflexion of the former, to the right of 1(^, having also D, I)' for inverse points.1008 The straight hnes drawn, from any point P, within or without a circle (Figs. 1 and 2), to the extremities of any passing through the inverse point Q, make equal chord angles with the diameter through FQ. Also, the four points QO QP. 0, A, B, P are concyclic, and QA QB
AB
.
=
.
therefore,fore, &c.
In eitlicr figure OR OA OQ and OR OB OQ (1000), by similar triantrles, Z OR A = OAR and ORB = OR A in figure But OAB = OBA (I. 5), there(1) and the supplement of it in figure (2). Pkoof.: :
:
:
Also, because Z
OR A = OR A,and therefore
the four points 0. A, B,(^.4.
P
lie
on a
circle in
each case
(III. 21),
QR = QO QR.
(III. 35, 3G).
1009
The inverseis
inversion
of a circle is a circle, and the centre of See the centre of simihtude of the two figures.
also (1087).be the point where the common PnoOF. In the figure of (l: CD.4 M.11,22: Me.62,,64,81 P.o2 Q. (J: TE.27,28: T1.21. ap. to linear complexes and congruences Me.83. ap. to tanf^ent of parabola: AJ. laelimination of afSy from the conditions of integrability of Suadp, &c. TE.27. equations C.98 linear, QQo of surJoachimstahl's method, faces,: : : : ::
Radii of curvature of a surface A.ll, 55: Q.12: Z.8. principal ones L.47,82 M.3 Me.80 N.55. *Radii of curvature of a surface 5795 5817 A.11,55 Q.12 Z.8. * ellipsoid 5831.:
:
:
:
:
:
:
:
:
:
*
L.48.2. flexible surface principal: 58146: L.47,82: Me.80: N.55.:
M.3:
constant*for
:
Me.64.:
:
;
:
an ellipsoid:
5832.:
E.43.
vpj.finitef.
TE.28::
qQ-qQ;
= 0,
AJ.4. for quantification of curves, surAJ.2. faces, and solids:
groups
geometry
of:
CD. 9.: :
integration ths Me.85. transformations Man.82.*Quetelet's curve: :
5249.
Quintic curves cnM.25. Quintic equations AJ.6i,7:
equal and of constant sign C.41 JP.21 L.46,,50. * Euler's theorem 5806. one a function of the other An. 65 C.84: J.62. product constant An. 57. reciprocal of product An. 52. sum constant An. 65. twice the normal C.42. sum Radius of curvature of a curve 5134 A.cn4,9,31,33 CD. 7 J.2,45 ths M.17: N.62,74: q.c and t.c Q.12: : : : :
:
=
:
:
:
:
:
:
:
An.65,682:
Z.3.
C.46..,48,50,61,62a,73,80.,85 M.13".,14,15: P.64: Pr.ll
J.59:
absolute * *:
:
CM.l.; : :: :
Q.3:
TI.19
:
Z.4.:
P-61 Q-3. condition of transformability into a recurrent form E.35.auxiliary eq. of=:
Man. 152
irreducible69.
:
AJ.7
ang. deviation, 5746. circular 5736 1259 A.9 CM.l J.30 of conies L.36 Me.66 Mel.2 N.45,682. N.54. at a cusp or inflexion point Me.81. in dipolar coordinates of evolutes in succession N.63.::
:
:
:
:
:
:
J. 34.:
functions of difference of roots:
An.
of
gauche curves:
:
l^.QQ.:
of a geodesic
L.44
on anratio(.'',
ellipsoid,
An.51.:
reduction of Q.6. resolvents of C.63o. whose I'oots ai'e functions of a variable: Q.5. solution of An.79::
**
and normal in constant of normal section of5817.:
:
N.44.
y, x)
=
:
J.59,87:
:
N.42
:
Q.
2,18.
Descarte