2IO On the RESOLUT'lON of 



But y = ^^+'"*^-'"'^' ; confequently, p'q'z—zp^q = 

 m^z+2?n^p~2mp\ znd z zz ^Jld±^f!l£=^tL . Now, j/ = 

 q'z~2qi whence by fubftitution, j = 2>«V?'-2>"£?;+a^>'y , 

 And becaufe x = 2^±?, we have alfo x = . 2^?'-2.;+2y ^ 



Cor. Let m ■=. i, then the formulas will be more fimple; 



X — 2A?'+ 2?— 2 ^ _ 2££^-2£V+2f, and z = -Mli±2£:i3£l 



There is a remarkable cafe in which the above formula do 

 not diredlly apply, the numerators and denominators vanifhing 

 at the fame time. It is when ni zz i, p ■=: 2, and q ■=. ^. For, 



by art. 3. y = ^"'^^ r^^'f +^"''^ = 2z2±l. = ^ ; where- 



fore the value of ^ may be expreffed by any aflumed number, n. 



But, by art. i. x = f^-^^L — y-^2 ; whence x = «+2. Alfo,' 



by art. 2. x zz ^±^ 1= ^±L ; therefore 2+4 =: 4«+8, and 



z zz 4«+4. Thus, 2, 4, 12; for 2X4+1 = 9, 2X12+1 = 25, 

 and 4X1 2+ 1 = 49. 



PROBLEM XIII. 



. 'to find a cube which Jhall be equal to the product of a fquare by a 

 given number. 



By hypothefis, x"^ =. ay^, and refolving, xXx'' =. aXy'^ ; 

 whence x = ma, and y'^ = mx'^ j but x'' = (''?^)% confequently, 



r 



