INDETERMINATE PROBLEMS. 19^ 



bitrary, and if rational, x and y rauft alfo be rational. Tranf- 

 pofing th« firft equation, x zz ma — y, and reducing the fe- 

 cond, mx — my zz b, and tranfpofing mx — b-\-my, and therefore, 



x zz — ; whence by equality — — ~ ma — y, and reducing, 



b-\-my — nfa — 7/iy, and tranfpofing 2my zz m 2 a — b, whence y zz 



tn x a — b i n i tn 1 b+a T n 



■\ but x zz ma — y, coniequently x zz . If m zz i ; 



2m J 2m 



, b+a , b— a 



then x zz , and y r: — — . 



Suppose it were required to find a number which, increafed 

 or diminiuied by io, would produce fquares. It is obvious, 

 that the number may be denoted, either by x 2 — io, or y 2 -\-io ; 

 whence x 2 — io — y 2 -\-io, and tranfpofing x 2 — y 2 = 5X4, and 



applying the above formula?, x zz 2m — ; if m zz 2, then x zz 6 



and the required number 26. 



PROBLEM II. 



To find two numbers , the fum of the fquares of which fhall be 

 equal to the fum of two given fquares. 



By hypothefis, x 2 -\-y 2 — a 2 -\-b 2 , and tranfpofing x 2 — a 2 zz 

 b 2 — y x t and, by refolving into factors, (x-\-a){x — a) zz 

 (H~jO {b—y) j whence, by fubftitution, x-\-a zz mb — my, and 



x — a zz — — . Tranfpofing the firft equation, x zz mb — my — aj 



reducing the fecond, mx — ma zh b-\-y, and tranfpofing, mx zz 



, , c ma+b+y ma+b+y 



ma-\-b-\-v, and therefore x zz : whence, rr: 



mb — my — a, and ma-\-b-\-y zz m 2 b — m 2 y — -ma, and tranf- 



b b 2 pofing 



