294 Mr. J. J. Sylvester on the 'Equation in 



form 9«+l, and in the second case ABC again of the same 

 form 9m ±1, but likewise D divisible by 9, j:? being in both 

 cases a prime, then the given equation will be generally inso- 

 luble. And I am now enabled to add that the only solution 

 of which it will in any case admit, is the solitary one found by 

 making two of the terms A.^■^, B?/^, Cz^ equal to one another ; 

 so that, for instance, if the given equation should be of the 

 form 



^ +y + AB C . a^ = D^ys, 



then the above conditions being satisfied, the one solitary so- 

 lution of which the equation can possibly admit, is J7= 1 ^=1, 



Az^- 02 + 2 = 0, 



which may or may not have possible roots. I call this a soli" 

 tary or singular solution, because it exists alone and no other 

 solution can be deduced from it ; whereas in general I shall 

 show that any one solution of the equation 



A*^ + B?/3 + C^^=Da3/0 



can be made to furnish an infinity of other solutions indepen- 

 dent of the one supposed given, /. e. not reducible thereto by 

 expelling a common factor from the new system of values of 

 x^y^ z deduced from the given system. 



The following is the Theorem of Derivation in question : 



Let 



A«3 + B^3^C/=D«/3y. 



Then if we write 



F=A«3 G=B/33 H=Cy3, 



and make 



.r=F2G + G2H + H2F-3FGH 

 j/=FG2 + GH2 + HF2-3FGH 



z=l{F3 + G3 + H3-3FGH}, 



or 



= «/3y{F2 + G2+H2-FG-FH-GH}, 



we shall have 



x^ -f- 2/3 -f. ABCs:^ = Yyxyz. 



I am hence enabled to show that whenever x^ -^-y^ -\- kz^ 

 = Djcyz is insoluble, there will be a whole family of allied 

 equations equally insoluble. For instance, because x^ + y^ + z^ 

 = is insoluble in integer numbers. I know likewise that 



x^ +y^ + z^=x^y^ + x^z^ + y^^ 



a^ + y*^ + a« = .ry + a^^ — 2^/3^3 

 are each equally insoluble. 



