ao8 On the RESOLUTION of 



Suppose a — 17, b — 6, c z=l 2, and let m = 2 ; then y 



I7+6—S 68+S—6 . 34+4+12 



= —— — = 3, x - — _ -= 14, and 2=— — ■ = 10. 

 But 14+3 = 17, and (14+6X3+2) = 100 = (io)\ 



PROBLEM XI. 



Let it be required to find two numbers , fuch that, if to each % 

 their fum and difference, unit be added, the numbers refulting fliall be 

 fquares. 



The firft condition will be obferved, if the numbers be de- 

 noted by x* — 1 andjy 2 — 1. The hypothefis will then require 

 K*-\-y* — 1 — z » f an( i ^»_j,*_|_ I = V 2 m 



Transposing the firft equation, x* — 1 -= z % —y % , and re- 

 folving into factors, (x-\-i)(x — 1) = (z+y)(z — y) ; whence, 

 x-\- 1 =: ?nz — my, and z-\-y = mx — m j therefore, z := mx — m — y 

 = ^Ltf±L from which we have _. *»*-«->"»-' . 



Again, tranfpofing the fecond equation, x % — y % = v 2 — 1, 

 and refolving, (x+y)(x—y) = (v+i)(v — 1), and by aflumption, 

 x-\-y r= pv — p, and v+i zz px — py, and therefore, v zz 



p x —py-i - J!±y±E- m Hence y = P 1 *-*-*!> . 

 r rJ p J p *+i 



But it was found, that y = m*x—x—m*—i w herefore, 



•^ 2OT 



r*-*-^ - «'«-*-"»-i . and by redudion, * 



p 1 + l 2m * J ' 



