562 [May 7, 



I annex two propositions, one qualitative, the other quantitative, 

 referring to such distributions. 



Qualitative Theorem. 



In any excyclic distribution between m urns o/* m-f-n 1 coun- 

 ters of n different colours, any set of counters selected at will 

 must be fewer in number than the number of distinct colours which 

 they contain added to the number of urns from which they are drawn. 



Before going on to enunciate the second proposition I must premise 

 one or two simple definitions. 



The capacity of an urn means the number of counters it will con- 

 tain, the frequency of a colour the number of counters of that colour, 

 so that the sum of all the capacities and the sum of all the frequencies 

 must be each equal to the number of the counters. 



Again, by the diminished capacity of any urn or diminished fre- 

 quency of any colour, I mean such capacity or frequency respectively 

 diminished by unity. 



Finally, by the polynomial function of any set of numbers a, b t . . . I, 

 I mean the coefficient of x a . y b . . . z l in the expansion of 



I can now enunciate the following 



Quantitative Theorem. 



The number of modes of excyclic distribution between m urns 

 q/" m + n 1 counters of n different colours is equal to the pro- 

 duct of the polynomial function of the diminished frequencies of all 

 the several colours multiplied by the polynomial function of the 

 diminished capacities of all the several urns. 



Observation. 



A double determinant means the resultant of a system of (m -f- n 1 ) 

 homogeneous equations each containing mn terms, and linear in respect 

 to each of two systems of m and n variables taken separately, but of 

 the second order in respect to the variables of these two systems taken 



collectively. Any such resultant is of the degree , n 



7r(m 1 )ir(n 1) 



in respect of the given coefficients, and may be represented by an ordi- 

 nary determinant of the (m+n l)th order, every one of whose terms 

 corresponds to a particular system of capacities of the m urns and of 

 repetitions of the n colours in the question above treated. 



