200 On the RESOLUTION of 



fing and dividing, x = - — — ; whence, — — r= -, and re- 

 ducing, m'^^-y — mf^ ■=. ey — meh, and tranfpofing, in'gy — ey = 

 fnfs — ^fh, and confequently, y z=. -~r-^- Alfo x = - = 



Suppose {7x-\-6){2x-\-\) =/». If /tz = 2, then « = -g^y- = a, 



and y = ^^^= 10 ; but 20X5 = 100 = f 10)'. 



Cor. Let e zz i, and ^ = i ; the hypothefis will become 

 i^+f) (*+^) = y'' In this cafe, we obtain x = ^^, and j = 



-^rzi- Thus, if (x+i2)(x+2), where / =: 12, and ^ = 2, 



12 -2- . 10 



and m = ^ J then x = i- =: 6, and y = — = 1 2 j but 



^-r -i— X 



4 4 



18X8 = 144= (l2)», 



PROBLEM VII. 



Let it be required to find rational values of x andy^ in the general 

 quadratic, hx^-\-^x-\-C z=. y'^. 



Case I. When the firjl term is a fquare. 



Suppose A ■=. «% when the expreffion becomes a'^x''-\-bx-\-c 

 — j' J by tranfpofition, bx-\-c — y — a'-x^, and refolving into 



fa<^ors, b[x-\- — j z: {y-\-(ix) [y — ax) ; whence, by afTumption, 



