INDETERMINATE A'NALYSIS. 



the folution of ttc equation .v^— y»= Cz' is always to be 

 obtained ; and in the following propofition, with whici: we 

 Ihall conclude tliis article, it will be feen that every equation 

 of the form .f' — Ay' = Bz', which is poffible, may be 

 transformed to another of the form x' — y'' = c s''; and that 

 X, y, and z, in the original equation, will be dependent upon 

 thofe of v', y', and %', in the transformed equation ; and 

 the'refore thcfe lull being known, the former will be known 

 alfo. 



Pkop. X [V. 



To tra'isform ever)- poffible equation of liie turm .»-' — 

 Ay- — B, to another dependent equation of tha form 

 x'^ _ _y" = f z<'. 



Ex. I. — It is reciuired to transform the equation .r' — 

 5 _v- =11 K", to another of tl»e form x' — v'* = c %". 



Having firtl afcertained the poflibillty of the equation by 

 the foregoing propofition, the transformation may be effect- 

 ed in the following manner. 



AiRime :c = n_y — i ly', and this fubllituted for x, gives 



/«' — <"' - 



Ex. 2. — Required the values of .r, jjandz, in the cqua' 

 tion 



Firfl, 

 e have 



13^'', and fubftituting for 



Znyf 



^ly 



Br; 



and here n = 5, whence the equation becomes 

 _)•" — 10 y y' + 13^''-:= z' ; or 



(y- ^y'Y — 12 y--^^'- 



Make now_)' — ^ y' = x', and it becomes 

 and the general values of .r' and z in this equation, art 



•'■■ - 3/- + 4? 

 Whence, by affuming /> = ; 

 a = 8, and ^' = 4 ; fo that 



3P 



and 



2pq. 



yy 



that is, let 



Take n, fo lliat n'' — ^ may be divifible by 

 7! =r 4, and our equation becomes 



/- iyy' + 1 1 J'" = -' ; '•"■ 



{y -4/)' - s y"' - =' 



or, by making^ — ^.y' = x, we have 

 x"-5y'-==z'; or, 

 x'-— a.' = ^y, as required, 

 that is, the equation has been reduced from the forr 

 5 v' = 1 1 z% to another of the form x'- — y" = c : 

 at leail, to x'- — z^ — § y'- ; which differs from the f 

 ing only in the letters. 



And, by means of the values of x, v', and 2, in this laft, 

 we readily arrive at thofe of x, y,- and 2, in the one propo- 

 fition ; 



for x" — y — 4_v'> or y= x' + 4j'' ■ 

 and X = ny — 1 1/, or 'x — ^^y— II y' 

 Now we have, feen, under the article Diophanttve, that 

 the general values of .,'' and 2, in eciiations of the form 



J' = -^ 



.r= 5 J 

 z =: z 



are as foUi 



sy'= 36 

 12/= 13 



-4?. 



md J = I, we haveV = i6, 

 the original equation, the va- 



I- I 



= 9 

 = 33 



orego- 



the latter values being formed by dividing the former by 

 their greateft common divifor ; and either of thefe fets of 

 numbers anfwer the required conditions ; for, 



132^-12. 36'= 13. 8, and 33=- 12.9'= 13.2'; 

 and various other values may be obtained by changing the 

 values of p and q. 



We will now give one example in wliich the required re- 

 duclion does not take place in the firft transformation, 



Ex. 3. — Required the values of .r, j, and 2, in the equa- 

 tion 



=z 191:. 



— 19^': then the fubftitulioa 



A ffume, as before, x = 

 of this value for x, gives 



yy'+^9y' 



Vvhence 



sr- i=.vy+ ^9y 

 ^sy - s-yy' + 95 y 



now, J ^ — JO y' = : 



- = z": in which n = ; 



:>^ 



Making now, j ^ — jo y' = a' ; we have 

 •»■ - 5 }" = 5 -■' 

 and here, though we have not arrived at the form requirec^ 

 the lad co-cfEcieiit is reduced from 19 to 5 ; and thus every 

 fucceilive transformation will reduce the co-cfScients, till we 

 ultimately arrive at that of unity ; but without purfuing this 

 reduction farther in the prefent Cafe, we are led to the folu- 

 tion in an eafier manner : for we fee immediately, that y may- 

 be alTumed = 5, ji' = 3, and 2=1. 



And from thefe we readily afccrtain thofe of x, y, and Zy~ 

 in the propoled equation ; for 



r _ -v' + ny' . _ 5 + »o-- a _ 



1 .V = ny — l<) y' ', Or .v = 50 — 38 = 12 

 Lz =: 1 ; or ^. = 1 . 



■which give 12^^-5 .5'' = I9'. J', as required. 



We fhall now conclude this article with a fyncpfis of in- 

 we mud again transform this anew, and, by continuing the determinate formu'se ; the dcnionilration and application of_ 

 operation, the reduftiofl to the flual feiui will be.ultimatelv which the reader will find in Barlow's " Elementary Invefti- 

 tiitttc^. gation.of the Properties of Numbers," 



Sjnopfn 



where p and q may be aiTumed any numbers at pleafure. If 

 / 2= 3, and ^ = I, we hdve x = 86, y = 38, and 2 = 4; 

 which numbers anfwer the required conditions ; for 



86* — 5 .38^ = II .4"-; 

 and by giving different values to p and q, a variety of other 

 integral values may be found for .v,^-, and 2. 



Note It . niiiy happen that the firfl transformation will 



not reduce the equation to the form required, in which caf( 



