380 Prof. Sylvester on a Generalization of a 



Thus if p = l, we have Cauchy's theorem, viz. 2 = 1 ; 

 ,, p=—\ } „ Cayley's theorem, viz. 2 = 0* ; 

 and in general for any value of p, 



p(p + l)...(p + n-l) f 

 l.2...n 

 In this theorem is in fact included another, viz. that if 

 ua + /3b+ ...+\l=n and » + /3+ .. . +\=r 

 (permutations not admissible), then 



« Un 



^n«.a a .n/9.^...nx.c^ 



is equal to the coefficient of p r ~ l in 



{p + 1)0> + 2>.,. 0>+»-i). 



This coefficient is accordingly (to return to Cauchy's theory of 

 arrangements) the number of substitutions of n elements capable 

 of being expressed by the product of r cyclical substitutions. As, 

 for instance, the number of substitutions of four elements a, b, c, d 

 capable of expression by the product of two cyclical substitutions 



* Provided, however, that n exceeds 1, a limitation accidentally omitted 

 in Mr. Cayley's paper ; and so in general 



2L-R1 =o, 



n« . a<*. . . n\ . ft 



p being any positive integer provided n is greater than p. 

 t If p=t> we obtain 



1.3. 5... (2«-l) 

 2_ 2.4.6... 2» ; 



from which it is easy to infer that the number of substitutions of 2n things 

 representable by the product of cyclical substitutions, all of an even order, 

 is (1 .3.5...(2«-l)) s . If p=-h we obtain 

 1.1.3. . .(2b.-1) 

 2_ 2.4.6...(2«) ' 

 combining which with the preceding result, it is easy to infer that the num- 

 ber of substitutions of 2« things representable by" the product of an odd 

 number of cyclical substitutions, all of an even order, is to the number of 

 such representable by the product of an even number of cyclical substitu- 

 tions, all of an even order, in the ratio of n to (n - 1 ). The former of these 

 two theorems is intimately allied with Mr. Cayley's celebrated theorem on 

 " skew," or what, for good reasons hereafter to be alleged, I should prefer 

 to call polar determinants, viz. that every such of the 2»th order is the 

 square of a Pfaffian. A Pfaffian is in fact a sum of quantities typifiable 

 completely, both as to sign and magnitude, by a duadic syntheme of 2» 

 elements, the number of which is readily seen to be 1.3.5...(2ra-l). I 

 believe I shall soon be in a condition to announce a remarkable exten- 

 sion of this theory to embrace the case of Polar Commutants and Hyper- 

 pfqffians. 



