INDETERMINATE PROBLEMS. 211 



y % zz w 3 a% and jXy = maXm*a ; and by a fecond aflumption, 

 y — pma } and m 2 a rz />y ; but x zz ma ; whence y zz px> and 



fince y z= _!L-f_ y zz — ; wherefore — = px, and x zz 



' p ap ap r 



ap 2 -, but y zz px, whence y zz ap*. 



Suppose a rr 3, and p zz 2, then x == 3X(2) 2 = 12, and 

 y =: 3.(2) 3 rr 24. For (12) 3 = 1728 = 3-(24) 2 . 



PROBLEM XIV. 



To find two numbers, the fum of which Jhall be a given fquare, 

 and the fum of their cubes afquare. 



By hypothefis, x-\-y 3E z*% and x 3 -j-y 3 ss 2 2 . Dividing the 

 fecond equation by the firft, we obtain !L- =: # 2 — *y +^% or 



i_ — -j 2 =t # 2 — xy, and refolving into factors, (-+j0 (-— Jf) = 



#(#— 7) y whence, x zz m{^ — y), and —-\-yzzm(x—y). By 



reducing the firft of thefe expreflions, z zz ax+ma y ; an d by 

 the fecond, z zz max — may — ay ; whence ax+ma y zz 



m 



max — may — ay. andy r= m x ~ x . But from the firft equation, 



m 1 +2m 



y zz a z — x > wherefore, m x ~ x =: a 1 - — x t and therefore x zz. 



m , +2w 



g*X „ » . „ — ; . But y = a? — x, confequently, y = a z X — ► 



d d 2 Alfo s 



