INDETERMINATE PROBLEMS. 197 



Suppose a 1= 22, b~ 3, and m =z 2, then x = a+6+i ' ^ ^^ 



= 14, and y = "1^^X22 =: 6. But 196+36+252 = 484 



= (22)\ 



Cor. If ^ — i, the hypothefis will be x''-{-j''-{-xy iz «* ; and 



2m+l m' — I , .- , 



* = r^.^+m+i ^'^' ^^^y = ^4:^TT^''- ^^"*' '^"^ = '3, and m 

 = 2, thcnx = -^:^^Xis=y,&ndy = -^^^X IS = 8. 

 But 49+64+56 = 169 = (I3)^ 



PROBLEM IV. 



To find two numbers, fuch, that each, increafed by unit, Jhall be a 

 fquare, and their /urn, increafed by unit, a given fquare. 



Let the numbers be denoted by x"" — i and j* — i, and 

 the firfl condition will be obferved. The laft requires, that 

 x'^ — i+j'' — i+i, or x^+j' — I ■=. «^ By tranfpofition, x' — i 

 = rt' — •/% and by refolution, (*+i)(x — i) zz {a-\-y){a—y) ■; 

 whence x+i — ma — ?;y, and mx — m zr a-\-y. Tranfpofing the 

 firft equation, x zz ma — my — i ; and tranfpofing the fecond, mx =- 



«+/+?«, and dividing, x zz. — ^^ -^ whence z=. 



ma — my — i, and reducing, a-\-y-\-m 1= m'^a — m'y — m, or 

 m''y-\-y zz m'^a — %m — a, and therefore y — — ^^13— — . But 



a+y+m m'^+2ma — i 



X ZZ — - — , whence x zz — -7— — . 



ru- ' m +-1 



Suppose 



