and Simultaneous Sign- successions. 465 



when n is a prime number, the system of (n — l) 2 equations 

 which give the terms of the nucleus admits of H(n — 1) . II (tz — 2) 

 systems of roots. 



It will be seen that this law does not hold when n is a compo- 

 site number, the rule for which I now proceed to state. 



6. (1) I observe that there will be as many distinct types of so- 

 lutions as there are distinct modes of breaking up n into factors*. 



(2) Let n=p . q .r . . . be one of the decompositions in ques- 

 tion. Write down the disjunctive product 



(l,a, a*,...aP- l Jl, b } b% 



b«- 



'X 1 



} c f c } 



.C 



X 



in which the terms are to follow any fixed law of succession. 

 This will produce a line containing p . q . r . . . , i. e. n terms. 



Let a, b, c, . . . respectively represent the pth, qth, rth, . . . 

 roots of unity ; by giving to each of these quantities successively 

 its p,q, r, . . . values we shall obtain p . q . r . . . , i. e. n lines, con- 

 stituting a matrix of the nth. order ; the totality of the matrices 

 so formed contain between them the complete solution of the 

 (».-— I) 2 system of equations. 



As an example let n = 4i. 



Here there are two modes of decomposition, viz. 



ft = 4, n = 2.2. 



Let i, i 1 denote the two primitive fourth roots of unity, and 

 denote negative unity by 1. The two types will be 



1111 

 1 i I i' 

 1-111 

 1 i' 1 t 



and 



1 



1 



l 



l 



1 



1 



l 



T 



1 



1 



I 



I 



1 



T 



I 



l 



The number of distinct derivatives of the nucleus of the first 



(1.2. 3) 2 



of these types is 



i. e. 18, the divisor 2 originating in 



the symmetry of the square in respect to its diagonals. 



The number of distinct derivatives of the second type, w T hich 

 contains a higher capacity of symmetry than the former (z. e. a 

 symmetry persistent under certain permutations of its consti- 

 tuent lines or columns), is 6. 



The following Table, in which + — are substituted for 1, I, 

 will make this evident. 



* When n is the vt\\ power of a prime, the number of decompositions 

 becomes the number of indefinite partitions of v. 



