On the equation x^ + j^ + As^zs'Man/z, Src, 467 



proved to be C,4 {q T O3J HO, or C14 ^Qf^ O3, HO, a 



dichlorobenzoic or a trichlorobenzoic acid, such existing. 



These experiments were conducted in the laboratories of 

 the Royal College of Chemistry under the direction of Dr. 

 Hofmann, to whom I beg to offer my best thanks for his ad- 

 vice and assistance during their progress. 



LXXI. On the General Solution {in certaiti cases) of the 

 equation a:^ + i^-{-Az^=Mxyz, i^-c. By J, J. Sylvester, 

 A.M., F.R.S., late Professor of Natural Philosophy in Uni- 

 versity College, London*. 



T SHALL restrict the enunciation of the proposition I am 

 -*- about to advance to much narrower limits than I believe 

 are necessary to the truth, with a view to avoid making any 

 statement which I may hereafter have occasion to modify. 

 Let us then suppose in the equation 



a^ -\-y^ -f- Az^ = Msyz 

 that A is a. prime number, and that 27 A — M^ is positive, but 

 exempt from positive prime factors of the form 6/+ 1. Then 

 I say, and have succeeded in demonstrating, that all the pos- 

 sible solutions in integer numbers of" the given equation may 

 be obtained by explicit processes from one particular solution 

 or system of values of x, y, z, which may be called the Primi- 

 tive system. 



This system of roots or of values of x, y, z is that system in 

 which the value of the greatest of the three terms x, y, A^.z 

 (which may be called the Dominant) is the least possible of all 

 such dominants. I believe that in general the system of the 

 least Dominant is identical with the system of the least Content, 

 meaning by the latter term the product of the three terms out 

 of which the Dominatit is elected. I proceed to show the law 

 of derivation. 



To express this simply, I must premise that I shall have to 

 employ such an expression as S'=f (S) to indicate, not that 

 a certain quantify, S', is a function of S, but that a certain 

 system of quantities disconnected from one another, denoted 

 by S', are severally functions of a certain other system of 

 quantities denoted by S; and, as usual, I shall denote <p<pS by 

 f^S, ffS by <p^S, and so forth. 



Let now P be the Primitive system of solution of the equa- 

 tion 



^ +y + A^^ = Mxyz, 



* Communicated by the Author. 

 2 H2 



