CURVILINEAR ASYMPTOTES 653

seeing them. The Science teacher who is really interested inteaching more than in entertaining should strive to make hisclasses realize that Science today is a group of principles, in-teresting it is true, but to be learned and mastered only byhonest effort on the part of the pupil himself.

NOTE ON CURVILINEAR ASYMPTOTES

BY LESTER DAWSONWicJiita, Kansas

Consider the curve whose equation is

F(x) m^W) (1)

where F(x) and f(x) are rational, integral algebraic functions of x andwhere the degree of F(x) exceeds the degree of/(.r) by an integer n^Q.We have by long division

y=aoxn-{-alxn~l-}- ’ - - 4-fln-i.r4-ffn4��- (2)/M

where r(x) is a rational, integral algebraic function of at least one degreelower than/(.r). Under the preceding conditions, the

limit^^O (3)A-^co f(x)

so that the curve (1) approaches more nearly the curve

y==00.r"+fll;t’n~l+ � ’ � 4-fln-l^+On (4)

as x increases indefinitely, in other words the curves (1) and (4) becomemore nearly congruent curves. The curve (4) may be termed a curvilinearasymptote of the given curve (1).Two important special cases yield ordinary linear asymptotes.Case I. If F(x) and/(.z*) are of the same degree, that is n =0, we obtain by

divisionr(x)

^^W)whence by (3), y^flo is the horizontal linear asymptote.

Case II. If the degree of F(x) exceeds that of/(.T) by one, that is n ==1,then equation (4) becomes

y=oo.r+oi

which is an oblique linear asymptote.It is clear that corresponding statements hold for the curve whose

equation is

-^ <5)f(y)

where F(y) and/(y) are subjected to the same conditions as are F(x) andf(x) in the preceding discussion.

654 SCHOOL SCIENCE AND MATHEMATICS

EXERCISES1. Show that y =a; is an asymptote of the curve

x^+Sx+ly�^r-’

^+^-32. The curve y =������has the parabolic asymptote y =x2.

x^+lx sin ^4-1 , ,

3. It is instructive to note that y=��\�� has the curvilinear

asymptote y =sin x, since by division y ==sin .T+� and since limit 1/x =0.x^oo

Hence in certain cases, the curve (1) may approach a curvilinear asymp-tote even when the restriction that F(x) and f(x) both be algebraic func-tions is removed.

STANFORD UNIVERSITY TO HOLD SUMMERCONFERENCE ON CURRICULUM

AND GUIDANCE

Stanford University will conduct a summer Conference on Curriculumand Guidance during the week of July 6 through 10 on the Stanford cam-pus. Among the nationally known educational leaders who will participatein the conference are John Studebaker, United States Commissioner ofEducation; George S. Counts of Teachers College, Columbia Universityand Research Director of the American Historical Association’s Commis-sion on the Social Studies; H. L. Caswell of George Peabody College, Nash-ville, Tennessee and Chief State Curriculum Consultant to the states ofVirginia, Alabama, Mississippi, Georgia, Florida, and Arkansas; C. L.Cushman, Director of Curriculum and Research of the Public Schools ofDenver; Worth McClure, Superintendent of Schools of Seattle; PeterSandiford, psychologist from Toronto University; C. A. Howard, VierlingKersey, and H. E. Hendrix, state superintendents of education of Oregon,California, and Arizona respectively; Frederick Redefer, Executive Secre-tary of the Progressive Education Association; R. D. Russell, CurriculumConsultant to the State of Idaho; and Emmett Brown, Professor of ScienceEducation at Teachers College, Columbia University and Lincoln Schoolof Teachers College. These and other visiting contributors will assist thestaff of Stanford University and the talent available in the public schoolsand colleges on the Pacific Coast.

This conference is planned to serve classroom teachers on all levels ofthe school system, curriculum workers, guidance workers, supervisors,administrators, research workers and the lay public.A stop-over at Palo Alto may be included in a round-trip rail ticket at

little or no extra cost for those attending the summer meeting of the Na-tional Education Association at Portland, Oregon. Details concerning thetotal program and the arrangements of the Stanford conference may beobtained by addressing an inquiry to Dean Grayson N. Kefauver of theSchool of Education, Stanford University, California.