1863.] 563 



The total number of such systems or terms will be 



J 7r(m+tt-2) I 2 

 l7r(m-l)7r(w-l)J ' 



Every term in this determinant will itself be a sum of simple deter- 

 minants of the (m+n l)th order, corresponding (each to each) with 

 the totality of the excylcic distributions of (m-\-n 1) counters 

 in respect of the particular systems of m capacities and n frequencies 

 appertaining to that term ; so that the number of simple deter- 

 minants whose sum constitutes a term in the grand total determinant 

 is always the product of two polynomial coefficients. In the par- 

 ticular case, where one of the systems contains only two variables, 

 one of these polynomial coefficients becomes unity, and the other 

 sinks down to a binomial coefficient. The only instance of a double 

 determinant which is believed to have been considered up to the pre- 

 sent moment is that given by Mr. Cayley in the ' Cambridge and 

 Dublin Mathematical Journal,' vol. ix. 1854, for the case of m=2, 

 w=2. 



IV. " On a Theorem relating to Polar Umbra." By J. J. SYL- 

 VESTER, M.A., F.R.S. Received April 27, 1863. 



By polar umbrae I mean such as obey in the strictest manner the 

 polar law of sign, so that not only any two appositions or products of 

 such umbrae derivable from one another by an interchange of two of 

 their elements are to be considered each as the negative of the other, 

 but also any such apposition or product becomes zero if the same 

 element is found in it more than once. 



Thus Sir W. Hamilton's t, j, k are not polar umbrae, because 

 although ijk=jik=kij> &c., ii,jj, kk, instead of being nulls, are 

 in the Calculus of Quaternions taken as unities*. 



Let us now define any set arranged either in line or column of such 

 umbral quantities to be multiplied by a corresponding set of actual 

 quantities when each term of the one set is multiplied by the corre- 

 sponding one of the other, and the sum taken of the products so 



* If we use Vandermonde's condensed notation for a determinant I"; o * ' ' n ~\ 



|_1 i . . . nj 



to represent a " determinant gauche," then, since on this supposition rssr and 

 rr=0, 1, 2, 3, . . . n will be polar umbrae by definition. 



