Numbers kx^ -\-'Qif -{■ Cz^ —T>xyz, 295 



In fact 



= t^ + x^ + li:^, 



where u, v, w are rational integral functions ofx,i/, z. 



Hence each of the factors must be incapable of becoming 

 zero*. 



As a particular instance of my general theory of transfor- 

 mation and elevation, take the equation 



x^ + y^ + 22^ = Mxi/z. 

 Then, with the exception of the singular or solitary solution 

 x=l .^=lj of which 1 take no account, I am able to affirm 

 that for all values of M between 7 and —6, both inclusive, 

 with the exception of M=— 2, the equation is insoluble in 

 integer numbers. 



Take now the equation where M= — 2, viz. 



x^+j/^ + 2z^+2xi/z = 0. 

 One particular solution of this is 



x=l y— — \ z = l. 

 Another, which I shall call the second f, is 



xszl j/=3 2=— 2. 



From the first solution I can deduce in succession the follow- 

 ing: 



j;=ll j/=5 z=— 7 



a?=— 793269121 3/=11794900O 2;= — 1189735855 

 &c. &c. &c. 



From the second, 



^=—10085 j/ = 8921 2;=— 844-2 

 a? = &c. y=&c. z = hc. 



As another example, take the equation 

 x^-\-Tl/^ + 6z^=6xyz. 

 One solution of the transformed equation 

 ^3 ^ 2v^ + Str* = Qumso 

 is evidently 



* It is however sufficiently evident from their intrinsic form, which may 

 be reduced to — (M^+SN^), that this impossibility exists for all the factors 



except the first, 

 t See Postscript. 



