SECTION III., 1902 RAS] Trans. R.S. C. 
XIX.—On a Theorem regarding Determinants with Polynomial Elements 
By W. H. Merzuns, B.A., Ph.D., FRS.E. 
Professor of Mathematics, Syracuse University. 
(Read May 27, 1902.) 
1. In a paper! entitled ‘On the development of determinants which 
have polynomial elements,” Dr. Muir gives the following three theorems : 
LAURE 
Let 2 p be a determinant of the nth order, each of whose elements 
consists of p terms ; let 2 oe p-l denote the sum of the p determinants 
formed from ne p by omitting, firstly, all the first terms of the elements, 
secondly, all the second terms, and so on; let 2 DA p-2 denote the sum 
of the 4 p (p-1) determinants formed by omitting, firstly, all the first 
and all the second terms of the elements of D, p: secondly, all the first 
) 1 
and all the third terms and so on, and let 2 pa = D 
PE, n, p-4, etc., 
bear similar interpretations, then 
Dre incl + Z LNA OO DE 0) RE) ee CE); 
ihe iT p denote the product of n p-termed expressions; if 2 IL, p-1 
2 1 
denote the sum of the p products formed from IT, p by omitting, firstly, 
all the first terms of the expressions ; secondly, all the second terms, and 
so on; if > We p-2, Z IL, Fo etc., bear similar interpretations ; then 
re en BGP ol CODEN SE oes (iI). 
The third is 
(a,ta,+.. 112) me (a +a, +... + de) + 2(a,ta,.+a 2) 
RE) EEE SO (DEN) de Monet's (ID). 
It will be seen that theorems II and III are particular cases of theo- 
rem I, 
For values of p 5 n the right-hand sides of these equations are no 
longer zero, and the principal object of this paper is to furnish their 
values under these conditions, ¢.e., to obtain general theorems true for all 
values of p. 
1 Messenger of Mathematics, New Series, No. 153, 1884. 
