no On the RESOLUTION of 



But y = g.»+*»V-*»»» . confequently, p*fz—*p*q = 

 m*z+2tn*p— 2«b/% and z = y'*+*»V-»"£l . Now, j/ = 

 tf-z—iq; whence by fubftitution, j = ^Vy'^V+^y . 



And becaufe * = J2±?, we have alfo x = , '^f . 

 Cor. Let ;/z rr i, then the formulae will be more fimplej 



x = 2/,?I + 2? ~ 2 y = 2/>y2 ~ 2 *y +2 ? - and.?— 2p 1 g+2p—2p l 

 p i q t -~i ' />V— * p x q 1 —* 



There is a remarkable cafe in which the above formulas do 



not directly apply, the numerators and denominators vanifhing 



at the fame time. It is when m rz i, p rr 2, and q z=. ^. For, 



by art. 3. y = a "V^ r ;^ m f +»"'? = 2l±tL = _£. ; where- 



J ° J p*q % —m % i_i o 



fore the value of y maybe expreffed by any affumed number, n. 

 But, by art. 1. x — 2m+y n j/-f-2 ; whence * =. /z-f-2. Alfo, 



by art. 2. * 5= Iffi. z= * +4 ; therefore z-f-4 =2 4^+8, and 



z =: 4«-f4- 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 pro duel of a fquare by a 

 given number. 



By hypothefis, x % =z ay 1 , and refolving, xXx* = aXy* j 

 whence x — /7z#, and j>* = mx 2 ; but #* = (ma)*, confequently, 



f 



