202 On the RESOLUTION of 



Case II. When the third term is a fquare. 



Suppose C zz. c^, and the exprefGon is ax'^-\-bx-\-c'' =:j/\ By 

 tranfpofition, ax'^-\-bx = j'^' — c% and by refolution, {ax-\-b')x — 



0'"l"0(j — '^) J whence by aflumption, x z= " — , and ax-\rb r: 



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



quently, JUl^H^l. ^y±L . whence j = ^fl^J^ilfL, and x = 



V+c lmc-\rb 



m "~" >n^ — a 



Suppose ^x''-\-^x-\-i6 =: j% and m ■=: 2 ; then x — 



-^ = 21, and y = -i^±i^ = 38. But 3.(21)^+5.21+16 



= 1444 = (38)^ 



Cor. I. Let b := o ; then the expreflion becomes ax^-]-c- 



= y\ and a; = ^—, Andy = —-;—-. Thus, 2x^+9 = j^ j 



if OT r: 2, .T =: -~- = 6, and j = -~i~ = 9- ^ut 2.(6)^+9 



= 81 = (9)S 



Cor. 2. If ^ = o, and c =1 i • then (3-t'+i := j% and .v = 



— ; — , and y 1= —z — . Put a zn m'' — d, and we fhall obtain x 



— -^, and y — ^ . Hence it is evident, that x and j' 



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

 Call the quotient « j then x = w, and y =: »z« — 1 3 whence 



— zz — ;« , or ?/i , which are the two firft terms 



of 



