296 On the Equation in Numbers Aa;^ + By^+C2;*D=a^a. 



Hence I can deduce an infinite series of solutions of the given 

 equation, of which the first in order of ascent will be 



a;=5 y=1 z-=S. 

 Again, the lowest possible solution in integers of the equation 



will be 



A-=17 .?/=37 z—'-^i. 

 The equation 



admits of the solutions 



37=1 j/=2 z=- — 1 



^'=—271 J/ = 919 2= -438. 



I trust that my readers will do me the justice to believe that 

 I am in possession of a strict demonstration of all that has been 

 here advanced without proof. Certain of the writer's friends 

 on the continent have, in their comments upon one of his 

 former papers which appeared in this Magazine, complimented 

 his powers of divination at the expense of his judgement, in 

 rather gratuitously assuming that the author of the Theory of 

 Elimination was unprovided with the demonstrations, which he 

 was too inert or too beset with worldly cares and distractions 

 to present to the public in a sufficiently digested form. The 

 proof of whatever has been here advanced exists not merely 

 as a conception of the author's mind, but fairly drawn out in 

 writing, and in a form fit for publication. 



P.S. It must not be supposed that the two primary or basic 

 solutions above given of the equation 



a^ -]-if -\- 2r^ + 2x1/ z = 0, 



viz. x = \ j/=— 1 z=l 



x=l t/=3 2;= — 2 



are independent of one another. The second may be derived 

 from the first, as 1 shall show in a future communication. In 

 fact there exist t/iree independent processes, by combining 

 which together, one particular solution may be made to give 

 rise to an infinite series of infinite series of infinite series 

 of correlated solutions, which it ma}' possibly be discovered 

 contain between them the general complete solution of the 

 equation 



x^-\-if^-ir kz^=\)xyz, J. J. S. 



26 Lincoln's Inn Fields, 

 Sept. 20, 1847. 



[To be continued.] 



