2o5 On the RESOLUTION of 



-J±L__ c; therefore * = 2 ^^**K To Amplify thefe 



formulae, put p rr z _ , and q rr ffl z__ a ; then will x zz ^ — cp, 



and j/ — />d — qac. If c become negative, the conditions of the 

 problem will not be affected. In this cafe, x rr pc-\-qd, and 

 y — pd-\-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^+7 — j/% then c z= i, and d — 3 ; and if ot 



— a, p =: -—^ rz 3, and q — —— zr 2 ; whence # zz 



3.1+2.3 zg 9 or 3, and > zz 3.3+4 = 13 or 5. Again, * = 

 3.9+2.13 zz S3 or h and y — 3- r 3+4-9 = 75 or 3- 0r * 



.== 3-3+2-5 = x 9 or x > and ^ = 3-5+4-3 = 2 7 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 — c y we {hall obtain for 



the exprefTion ax* — 1 zz y\ x zz icd, and/ zz d* + ac % . 



Thus, in the example, 2x 2 — 1 zz j/% where r zz 2, and d zz 3, 



x — 2.2.3 — I2 » a^ y — 3-3+ 2 - 2,2 — J 7 J and a S am > * — 

 2.12.17 zz 408, and y zz 17. 17+2. 12.12 zz 577. 



PROBLEM IX. 



To find two rational numbers, the fum of which Jhall be equal 

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



By hypothefis, x-\-y zz a, and x 2 +j 2 zz %\ Tranfpofing the 

 fecond equation, x* zz z 1 — y 2 , and refolving into factors, xXx 



