200 On the RESOLUTION of 



iing and dividing, x n y ~~ m ; whence, " — -~~ m ^ and re- 

 ducing, m^gy — mfg — : ey — meh, and tranfpofing, m % gy — ey = 

 ^ — ^, and confequently, y = —7-^7. Alfo * = ~~~ — 



Suppose (jx-\-6)[2x-\-i) =zy*. limzzi, then x zz 8 = a, 



and j = -fEr = 10 ; but 20X5 = 100 = (io)'. 



Cor. Let e — 1, and ^ z 1; the hypothefis will become 

 (*+/) (*+^) = J'*- In this cafe, we obtain x = "^Zl* and j/ = 



m z_ x . Thus, if (#-f-i2)(#-f2), where / = 12, and b = 2, 



12 -2_ . 10 



and »z r= — ; then a? s i. = 6, and y =: — = 12; but 



-2. -x -i— x 



4 4 



18X8 = 144= (l2)\ 



PROBLEM VII. 



Let it be required to find rational values of x andy, in tbe general 

 quadratic, Ax 2 -\-Bx-\-C = y*. 



Case I. When the firjl term is a fquare. 



Suppose A zz a z y when the expreffion becomes a 1 x' 1 -\-bx-\-c 

 zz y* $ by tranfpofition, bx-\-c — y 2 — a*x 2 , and refolving into 



factors, b(x+ —J zz (y-\-ax) (y — ax) \ whence, by alTumption, 



x+ 



