VARIATION. 



4 : -^; orA 



b : B. 



to a, B is chxiged to i, in fuch a manner, thnt 



1 I 



B ' J 



For example, if the area of a triangle be given, the bafe 

 varies inverfely as the perpendicular altitude ; for let A and 

 a reprefent the altitude of two triangles of equal areas, and 

 B and i their two bafes ; then 



A X B a X i 



or A X B = a X i ■■ 



i : B ; or A 



2 2 



therefore, 



I I 



3. One quantity is faid to vary as two others jointly, if, 

 when the former is changed in any manner, the product pf 

 the other two is changed in the fame proportion : that is, 

 A varies as B and C jointly, or A cc B C, when A cannot 

 be changed to a, but B C is changed to be, fuch that 

 A : a :: BC : be. The area of a triangle, for example, 

 varies as the bafe and altitude jointly ; for let A, P, B, re- 

 prefent the area, perpendicular, and bafe of one triangle, and 

 a,p,b, the correfponding quantities in another; we know 



, A PB 



that A = •§ P B, and a = i/i ; confequently — = — , 



or A : a :: BP : i/.. 



4. One quantity is faid to vary diredly as a fecond, and 

 inverfely as a third, when the firft cannot be changed in any 

 manner ; but that the fecond, multiplied by the reciprocal of 

 the third, is changed in the fame proportion. That is, A 



vaiies as -z^, or A 0= 



and a, b, c, being correfponding values of thefe quantities. 



For example, the bafe of a triangle varies as the area 



direftly, and as the altitude inverfely ; for as in the pre- 



■O p A 



ceding example, -7— = — ; if wt multiply both fides by 



t, we have y = ^^, whence B : b :: y : j. 



The following are fome of the principal propofitions re- 

 lating to the ratio of variable quantities. 



If A ex B, and B ex C ; then A oc C. 



If A « B, and B cc I.; then A oc ~. 



If A oc C, and B oc C ; then A + Boc^BAocC. 

 If A or B, and m is any given number, A oc m B. 

 If A or B ; then A" oc B", or A'^ oc B". 

 If A oc a, and M oc m ; then A M oc a m. 



■— , when A : 



J:-;A,B,C, 



K^ C 



If A a BC 



A A 



then B a -^ , and C 00 ^. 



ization far beyond what it was fuppofed capable of poirefT- 

 ing. This method was alfo in the interval illufl rated in 

 the moft fimple and elementary manner by the celebrated 

 Euler, in the Memoirs of the Academy of Sciences at 

 Peterfburgh for 1764, as it was afterwards in the third 

 volume of his Calcul Integral, and again in the Afta Petro. 

 for 1 77 1. Since that time it has IJcen treated of by dif- 

 ferent authors at greater or lefs extent ; and to Mr. Wood- 

 houfe, of Cambridge, we are indebted for a very neat little 

 volume, in which this fubjeft is handled in a very clear and 

 confpicuous manner, from which work we have already 

 . given a few extrafts under the article Isoperimetry. 

 BoiTut, alfo, in vol. ii. of his " Traites de Calcul DifFerentiel 

 et de Calcul Integral," has a very perfpicuous chapter on 

 the calculus of variations, of which we (hall avail ourfelves 

 in the prefent inilance. 



Let there be any indefinite expreflion or funtlion com- 

 pounded of variable and conflant quantities, which changes 

 its value by the increafe or diminution of one or more of 

 the elements which it contains : it will tlius undergo a 

 variation, and the method of finding this is what is called the 

 calculus of •variations. 



In the fame manner as x is made to denote the fluxion of 

 X, and d x the differential of x ; f o 5 .v is ufed to indicate 

 the variation of x ; and the fundamental rules of this calculus 

 are founded on the fame principles as thofe of the differ- 

 ential calculus : at the fame time, however, it is neceffary to 

 guard againfl confounding the one with the other. A very 

 fimple example will fhew clearly the diftinftion that mutt 

 be made between the two cales. 



Let us fuppofe the equation ji ^ = a k, which denotes the 

 relation between the abfcifs A P = *, and ordinate F M=y, 

 of a parabola AM {Plate Xlll. Analyfis, Rg. i.), a being 

 the parameter. By drawing p m indefinitely near to P M, 

 and M r parallel to the axis A V ; the line Y p, or M r, 

 will reprefent the differential d x, and M r the differential 

 d V ; and the relation of thefe differentials is found by the 

 differenciation of the equation y'' ■= ax, which gives 



a d X adiX 



2ydy = ad*, ordv = > or d v = . 



■^ ^ •' 2JI •'2 s.' ax 



Let us conceive now that the equation y'^ z=a x, vary by 

 the indefinitely fmall augmentation of its parameter a, wflich 

 is one of its elements ; and let us conflruft a fecond parabola 

 A N, which has a -)- Ja for its parameter. Then fuppofing 

 the abfcifs A P to continue the fame for both parabolas, it 

 is obvious that the ordinate P N, of the parabola A N, will 

 have for its value the primitive ordinate P M, augmented 

 by the fmall quantity M N, which therefore reprefents the 

 variation that the ordinate P M undergoes in confequence 

 of the variation of the parameter a ; hence, in reprefenting 

 hy ^ y the variation oi y, as that of a is denoted by a -f- ^ a, 

 the new equation will be [y -\- ^ y)' := (a -|- Ja) k ; from 

 which fubtrafting the original equation y'' ■= ax, we fhall 

 have (neglefting, as in the differential calculus, the variations 



I 

 A' 



If A B be conflant ; then A oc -— , and B « 



a 



If A Of B, andC oc,D; then A C « B D. 



Wood's Algebra. 

 Variation, Calculus of, is a department of the modern 

 analyfis, which we owe, as a diflinft branch, to the inventive 

 genius of Lagrange, who publifhed his firft memoir on this 

 fubjcA in the fecond volume of the Tranfaftions of the 

 Academy of Sciences of Turin, in 1762 ; and his fecond 

 memoir, publifhed in the fourth volume of the fame Tranfac- 

 tions, in 1 770, gave to this theory a perfection and general- 



of the fecond order) 2yly = xoa, OT^y=:- 



2> 



ly 



= ■, an equation which exhibits the relation of the 



2 ^ax 



variations }a and ^y. 



If, alfo, we make the abfcifs A P vary by the indefinitely 

 fmall quantity V p =■ ^ x, the correfponding ordinate for 

 the parabola A N will be qn, and the line s n will repre- 

 fent the variation of the primitive ordinate P M. Now to 

 find the relation between the variations J a, ^ x, ^y, vtc- 

 muft fubftitutc in the equation y'^ — ax, (a + i a) for a, 



