444 Prof. Cayley on the Mathematical Theory of Isomers, 



By addition of (A) and (B), 



« 2 = «? + /3J, 



which, on writing x* for x, is 18. 



We are now in the position to prove 19 ; for 



(l+2x + 2x 4 + ...) 4 -(l-2x + 2x 4 -...) 4 



= {( a + ^+( a -^}{(« + ^-(a-/3) 2 } 



= 8 fa +&)*»$ by (A) 



= 16{l+2x* + Za*+... }*{*?$ + J 1 + x^ + ...}* i 



on substituting for a,, /3 X their values, and using 17 with x 4 

 written for x, which transforms ^<x/3 into the second squared 

 factor, 



= l6(x$ + x* + x^ +. . .) 4 , 



by applying 17 again, x 2 being therein written for x ; and the 

 formula 19 is thus established by elementary algebra in a most 

 simple and elegant manner. 



The formula 18 is not required in the proof, but was no 

 doubt added by Gauss for the sake of completeness. 



Trinity College, Cambridge, 

 April 25, 1874. 



LVII. On the Mathematical Theory of Isomers, 

 By Professor Cayley, F.R.S* 



I CONSIDER a " diagram," viz. a set of points H, 0, N, C, 

 &c. (any number of each), connected by links into a single 

 assemblage under the condition that through each H there passes 

 not more than one link, through each not more than two 

 links, through each N not more than three links, through each 

 C not more than four links. Of course through every point 

 there passes at least one link, or the points would not be con- 

 nected into a single assemblage. 



In such a diagram each point having its full number of links 

 is saturate, or nilvalent : in particular each point H is saturate. 

 A point not having its full number of links is univalent, biva- 

 lent, or trivalent, according as it wants one, two, or three of its 

 full number of links. If every point is saturate the diagram is 

 saturate, or nilvalent; or, say, it is a "plerogram;" but if the 

 diagram is susceptible of n more links, then it is w-valent ; viz, 

 the valency of the diagram is the sum of the valencies of the 

 component points. 



Since each H is connected by a single link (and therefore to 

 * Communicated by the Author. 



