INDETERMINATE PROBLEMS. 211 



X" — m^a\ and yXy — maXm^a ; and by a fecond aflumption, 

 y — pma, and iri'a = py ; but x = ma ; whence y = px, and 



fince y = — — , y = — ; wherefore — = px, and x = 

 ap'^ ; but / = /x, whence y ■=. ap^. 



Suppose ^ = 3, and p — 1^ then A? = 3X(2)^ = 12, and 

 J = 3.(2)^ =: 24. For (12)' = 1728 = 3-(24)\ 



PROBLEM XIV. 



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

 and the fum of their cubes afquare. 



By hypothefis, x-\-y = «% and x^-\-y^ = z*. Dividing the 

 fecond equation by the firft, we obtain 5_ r: x'^ — xy +/', or 



21 — -j' = X'' — xy, and refolving into fadors, (7+j) (7 —J') = 

 x{x—^y); whence, x = ;«f^ — y), and — +>' = ffzOr— j/). By 

 reducing the firft of thefe expreffions, z = ''x+'"''y . ^^^ ^ 

 the fecond, z = OT«,t — ^ffz^j — ay ; whence "" . '^"'"y =z 



m 



max — 7nay — ay, and/ = »'^«— » ^ g^. fj-Qj^ the firft equation. 



y =: <3' — X ; wherefore, "" " " zz a" — x, and therefore x == 

 a'X a„,,^^^_^ . But J/ = rt^— Af.confequently, / = «*X 



+ 2/3— I •> » ^ ■' ' ^ 2»J'+2OT— I 



d d 1. Alfo^ 



