202 On the RESOLUTION of 



Case II. When the third term is afquare. 



Suppose G zz c % , and the expreflion is ax*-\-bx-\-c* zz j/\ By 

 tranfpoiition, ax*-\-bx zz y z — c\ and by refolution, (ax-\-b)x zz 



(y+c){y — c) j whence by aflumption, x zz * — , and ax-\-b zz 



my — inc. But from the fecond equation, x zz my , confe- 



quently, «±±. = J £ ; whence , = =*£fc, and * = 



y+c lmc J rb 



m m — a 



Suppose ^x z -{-^x^-i6 zz y, and /# zz 2 ; then x zz ■ 



■^f. zz 21, and j = ' 6 ^» zz 38. But 3 .( 2I )^ 5 . 2I+l6 



zz 1444 zz (38)*. 



Cor. 1. Let 3 zz o ; then the expreffion becomes ax*-\-c 2 



zz j/% and x zz -r— , and ;/ zz — 73^-. Thus, 2*S-o zz ^ ; 



if m zz 2, x zz -~- zz 6, and j zz ~~~ = 9. But 2.(6) 2 +o 



zz 81 zz ( 9 >. 



Cor. 2. If b zz o, and c zz 1 ; then ax*r{-i zz j 2 , and * zz 



— : — , and y zz — ; — . Put a zz wz 2 — d, and we fhall obtain x 



m — a J m — a 



zz — , and y zz — ~ d . Hence it is evident, that x and y 



will be exprefTed in whole numbers, when 2m is divifible by d. 

 Call the quotient n ; then x zz n, and y zz mn — 1 j whence 



y mn — 1 1 d _ ~. _ '-. : . 



— zz zz m , or m — — -, which are the two rirft terms 



x n n ' 2m ' 



of 



