INDETERMINATE PROBLEMS. I0^ 



bitrary, and if rational, x and/ mufl alfo be rational. Tranf- 

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

 cond, mx — my = b, and tranfpofing tnx zz: b-{-my, and therefore, 



X — ■ "^'"-'' ; whence by equality "*""''' rr ma — -)', and reducing, 



b-\-7ny zz m''a — my, and tranfpofing 2my z=. m'^a — /;, whence y zz. 



"* ''~ — ; but X = ma^y, confequently x =z - — —, If m zz i ; 



then X — , and y ■=. — —. 



Suppose it were required to find a number which, increafed 

 or diminifhed by lo, would produce fquares. It is obvious, 

 that the number may be denoted, either by x^ — lo, or j''+io ; 

 whence x'— lo r=. j'+io, and tranfpofing x' — -v' = 5X4, and 



applying the above formulae, x zz. — —^ — ; \i mzzi, then x zz-d 



and the required number 26. 



PROBLEM II. 



7<5 find two numbers, the fum of the fquares of which pall be 

 equal to the fum of two given fquares. 



By hypothefis, x'+j^ =r rt'+^S and tranfpofing x"^ — ^' ■=. 

 b^ — y"-, and, by refolving into fadors, {x+a){x — a) — 

 [b-\-y){b—y'); whence, by fubftitution, x+a — mb — my, and 



X — a zz — —. Tranfpofing the firft equation, x = mb^my — Ui 



reducing the fecond, mx — ma ±: 3+j/, and tranfpofing, mx zz. 



ma+b+y ma+b +y 



ma-\-b-\-y, and therefore x =■ — 3 v?hence, — — — 



mb — my — a, and ma-\-b-\-y = m'-b — m'^y — ma, and tranf- 



bb ^ pofing 



