212 On the RESOLUTION, &c. 



AHb.becaufe z =-^^:t^^, we have by fubftitutlon, z = 



m 



Cor. I. If a =: 2m''-\-zm — i, two whole numbers maybe 

 always found, the fum of which, and that of their cubes, fliall 

 be fquaies. For in this cafe, x =z (2ot'+2OT — i)(?«'+2ot), y = 

 (2OT^+2;« — l)(»i* — i), and z = (2m^-^-2m — i)"(»2^+?«+i.) 



Thus, if m = 2, we fhall find x = 8S, y z=. 33, and 2 = 



847. But 88+33 = 121 = (ii)% and (88)^+(33)' = 717409 



= (847)^ 



Cor. 2. If / be negative, we (hall obtain two numbers, the 



difference of which, and that of their cubes, fhall be fquares. 



Put m = -^, and fubftituting, x =: a^X ■—- — ~- , y = 



1 _£_ + _£-_i 



r 1 ~ 



dudlion, X - a-^X — f '"^^^^ . , y - a'X-^^-f—, , and z = 



' 2p'+2pq — q- ' •' 2p-+2pg — q' 



^3v P^+M+f . . \i a -zz 2p''-\-2pq — j% we fhall obtain 



2p''+2pq — y" 



whole numbers J for x = {2p'+2pq—q'-)(p^+2pq), y = 



i2p^-+2pq—q'){q'—p% and Z = (2/.'+2/>?— f )'(/)^+/i?+f ). 



These examples will probably be thought fufficient to ex- 

 plain the application of this method to the folution of indeter- 

 minate problems in general, and to fhew that it is not lefs ex- 

 tenfive, and much more uniform, than thofe that are commonly 

 in ufe. 



XV. 



