19 6 On the RESOLUTION of 



pofing ?n 2 y-\-y — m 2 b — zma — b, that, is, y — 



m 1 b — 2ma — b 



— m * +l — • But x zz mb — my — a, and fubftituting, x zz 



m 2 a+2mb — a . 



— m *+i — ' Thus, if a = 5, and b zz 10, and m zz 2 ', then 



4-IO— 4.5— 10 4.C+4.IO— « - / X , 



J = j- J = 2,, and x = ^^-^ 1 ~ 1 1 5 but (1 i) a -f 



(2)* = 125 = (io)»+(5) a . 



Cor. If £ == o, we fhall obtain two fquares, the fum of which 



fhall be a given fquare. For y zz — //2 * +I > or ~^~ m % +i > anc * * = 



#2*0 — a . 4.10 



— t . T . Thus, if # == 10, and m zz 2. then y zz — - — n 8, and 

 * = —— - — = 6, but 64+36 = 100. 



PROBLEM III. 



To find two rational numbers, the fquares of which, together with 

 any given multiple of their produtl, fhall be equal to a given fquare. 



By hypothecs, x 2 -\-y 2 -\-bxy =z a*, and tranfpofing x 2 -j-bxy =z 

 a 2 — y 2 , and refolving into factors, x{x-\-by) n (*+j0(* — y) ; 



whence, by aflumption, x-\-by zz ma — my, and x — . 



Tranfpofing the firft equation, x zz ma — my — by ; confequent- 

 ly, — — zz ma — my — by, or a-\-y zz m 2 a — my — mby, and again 

 by tranfpofing, m 2 y-\-mby-\-y zz m 2 a — a > whence y — 



m* — 1 a+y 2m+b 



— , , ,. Xa. But * = —- — .wherefore x =: — ,. ,. Xa. 



Suppose 



