212 On the RESOLUTION, &c. 



Alfb, becaufe % zz ax+ma y we have by fubftitution, z == 



m 

 2?» l -f-2»z — i 



Cor. i. If a zz zmP-^-im — i, two whole numbers maybe 

 always found, the fum of which, and that of their cubes, (hall 

 be fquares. For in this cafe, x tz {2m 2 -\-2m — i)(jn 2 -j-2m), y zz 

 (2m*-\-2tn — i)(m 2 — i), and z =: [2in 2 -\-2m — i) 2 (//z 2 -f 7/z-f-i.) 



Thus, if m = 2, we fhall find x — 88, y zz S3* an< * 2 zz 

 847- But 88+33 = 121 = (n)\ and (88) J -f (33) 3 = 717409 



= (847) 2 - 



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



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



Put m = -£-, and fubftituting, x zz a 2 X —r — — - — , y zz 



q* q - 



~~* % X 2 J1 2L_ » and * ~ a ** ltl k_ t ' and by rC " 



dudion, x = a 2 X — tl±m — v — a 2 X - f p * , , and 2 = 



» 2p'+2pq—q 2 ' ^ 2p*+2pq—q* ' 



a?X J X+J, ?± 3 — . If tf = 2p 2 +2pq—q 2 y we fhall obtain 



whole numbers j for x = (2p 2 -\-2pq — q*)(p 2 -{-2pq), y zz 

 (2p 2 +2pq — q i ){q z —p z ) i and 2; = (2p 2 -\-2pq — q' 1 ) t (p' l -\-pq+q' 1 )' 



These examples will probably be thought fufEcient 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. 



