io5 On the RESOLUTION of 



-c ; therefore x = 3:!!!tC^l±l)i. To fimplify thefe 



V+1 



formulas, put p zz ,_ , and q = T^JZ^ > then will x r; dq — cp, 



and y zz pd — qac. If c become negative, the conditions of the 

 problem will not be afFeded. In this cafe, x zz pc-\-qd, and 

 y :=. p(1-\-aqc. The values of x and y, obtained from either 

 of thefe formulae, may be repeatedly fubftituted for thofe of 

 c and d ; and thus a variety of numbers will be difcovered. 

 Suppose 2.1-^+7 ^=- j'% then c zz i, and J — 3 ; and if m 



— 2, p — —^ zz 3, and q zz zi 2 ; whence x zz 



3.1+2.3 = 9 or 3, and y zz 3.3+4 zz 13 or 5. Again, x zz 

 3.9+2.13 = 53 or I, and y zz 3.13+4.9 zz 75 or 3. Or x 

 = 3-3+2.5 = 19 or i> and/ = 3.5+4.3 = 27 or 3; and fo 

 repeatedly. 



We may obferve, that the value of p is the fame with that 

 of y in Prob. VI. Cor. II. and the value of q the fame with 

 that of X. Whence, if p zz d, and q zz c, we (liall obtain for 

 the expreflion ax'' — i = j'% x = 2cd, and y zz d'^ -\- aC. 

 Thus, in the example, 2X"' — i zz y"-, where c zz 2, and d zz ■^, 

 X zz 2.2.3 — '2, and y zz 3.3+2.2.2 = 17 ; and again, x zz 

 2.12.17 zz 408, and J =: 17. 17+2. 12. 12 =; 577. 



PROBLEM IX. 



To find two rational numbers, the fiim of which fhall be equal 

 to a given number, and the fum of their fquares afquare. 



By hypothefis, x-{-y zz a, and .t'+j" zz z\ Tranfpofing the 

 lecond equation, x' zz z' — y-, and refolving into fadors, xXx 



