INDETERMmATE ANALYSIS. 



denominator conlills of two prime faftors, 

 will then be to fiiid 



1'' 



a b, it 



•f integral folutions. If the gi^-en fratlion be 



;n whicli equation, having determined the values of p and q, 



ve fhall have r y for the fradions required ; and as 



many different ways may any fuch fradion be decompofed 

 into two others, as the equation a q + b p z^ m admits 



-, then 



we may firft refolve it into two fratitions, and one of thcfe 

 into two others ; thus, let 



'" ^ / . ? ■ 

 abc ab ' c ' 



then we have 



al q + c p = tn; 

 and having, from this equation, found the values oi p aT>d q, 

 P , 9 



• (hall have 



abc 



ab 



Affair., lot -i- = — 4- — , or E 

 aba 



J in this equation, fo (hall we have 



r b = p, find r and 



a b c a b c 



as required ; and in the fame manner may any fraftion that 

 admits of decompofition be refolved into others of which 

 the fun. (hall be equal to the original fraftion. 



Ex. — What are thofe two fraclions whofe fum is equ;d 

 to 1-9 

 35 



\ve may make — 

 Z5 



Since 35 = 

 produces this equation, 



5/ + 7 7= 19 

 in which the value oi p — i, and q = 



and therefore the fractions fought are 



readily found : 



2 19 



Pnop. VII. 



To find the leafl; number, that, being divided by given 

 numbers, (liall leave given remainders. 



Let N reprefcMt the required number, fuch that, being 

 divided by a, a', a" &c. the remainders- (liall be refpedtively 

 l, b', b", &c. that is, 



ti = am-^b — a'n + b'= a' p + B" &c. 

 and it is required to find the leaft value of N, that anfwcrs 

 thefe conditions. 



Firft fmce a m + b = a' n J^ b' 



we have am — a' n = b' '— b. 



Find, therefore, in this equation, the leaft values of m and 

 n by Prop. II., then will a in + b, or a' n -t- b', exprefs the 

 lca(i number that fulfils the firll two conditions. Let now 

 this number be called c ; then it is evident that every num- 

 ber of the form a a' q + c will alfo fulfil thefe conditions ; 

 and we muft proceed to find a a' y + c = a" p \- b" ; or, 



a c{ q — a' p — b" — c ; 

 that is, the leaft value of /> and q in this equation ; fo (hall 

 ■we have, a a'.q + e for tne . leaft niynber that anfwers the 



firft three conditions, and fo on for as many otliors as may 

 be propofed. 



Ex. — Find the leaft number, that, being divided by 28, 

 19, and 15, fhall leave for remainders refpeftively 19, 15, 

 and II. 



Here we have 28 ;n + 19 = 19 « + I5 = 15 /> + H- 

 Now, in the equation 28 m — 19 n = — 4, the leaft values 

 of m and n are m — 8, and n = 12 ; whence 28 m + 19 = 

 19 « -;- 15 =: 243 ; and it now remains to find 28 . 19 y 4- 

 243 = 15 /> + 11; or, 532 q- i;p= - 228. 



In which equation p = 512, and q = 14, whence 

 532 5' + 24; = ij ^ 4- II = 7691, which is the leaft 

 number having the required conditions. 



Having thus treated of the folution of indeterminate equa- 

 tions of the firft degree, to as great an extent as our 

 limit will admit of, we (hall proceed to thofe, in which one, 

 at leaft, of the unknown quantities enter in the fquare power, 

 which conititute the clafs of indeterminate equations of the 

 fecond degree ; and as we have feen that every equation of 

 the firft clafs lias its folution depending \ipon that of the 

 equation a p — b q ^ +1; foin thofe that we are about 

 to iuveftigate, the folution is intimately connefted with that 



ofti, 



(hull firft I 



: equation p 



N 



I ; this, therefore, is what 



Prop. VIIL 



To find liie integral values of / and j in the eqv.ation 

 /'- N?'= + 1, 

 N being any given number whatever, not a complete, 

 fquare. 



In order to obtain the general folution of this equation, 

 which is always poffible (at leaft with the pofitive fign) we 

 muft (hew the method of extrafting the fquare root of 

 any number N, not a complete fquare, in continued frac- 

 tions ; but as this operation belongs properly to the latter 

 fubjeft, we (hall only in this place indicate the method, and 

 mull refer the reader for the demonftration to vol. ii. of 

 Euler's Algebra, and to Barlow's Elementary Inveftiga- 

 tions. See alfo SQUARE Root. 



The transformation of the ,/ N, to continued fraftions, 

 is performed by means of the following formula. 



Let a be the greateft integer contained in .y'' N ; thea 

 make 



N - m= 



&c. &c. &c. &c. 



In thefe formulre a, u', u", &c. are the greateft integers con- 

 tained in the correfponding fraftions, which quantities are 

 the quotients, whence the converging fraftions are to be 

 deduced, by the fame rule as is given in Prop. I. for the 

 quutients a, fi, 7, ^, &c. : and by continuing tlie above ope- 

 rations, we ftiall be finally led to a quotient equal to 2 a, 

 at which term we muft Hop, and the correfponding fraftions 

 to the laft quotient before this will give the required value; 

 of p and q. 



Ex. 



