[merzcer] DETERMINANTS WITH POLYNOMIAL ELEMENTS 159 
for p = 2 and n = 3 








ay + Oy Ay + by» Az + Vus Ay dy Ars | by Oy Os 
Ae, + da A + Do» As + Bog} — | Ax, G22 A5| — | by De Doe 
as, + Va Ag, + Doo Ag, + Ogg dy dy Asg a Vg by 
| 
Ay, A, Oy Ay Oy ay bi Ay, dy 
— Pa heater Oa | o> || Gay boy Go| > | Oa Gz A» 
| | 
As, Az Ogg As, Vs y by y gg | 
| nl 
Ay Op Vis | by A. Ors bi Py Gs | 
+ lan Ox ag | + | by a» dm | Bay Onn ag 
As, Os by | Ds, y Ogg by ge sg 





3. If we make all the terms vanish in all the elements of 4 except 
those in the principal diagonal, it reduces to the product of n polynomials, 
and if {7 ay bg Pr denote the product of a a’s, 8 b’s, y c's... 7 p's, 
then theorem (B) becomes 
Te ee ae ch Ce 
Se el tate pe ee (C) 
When p>n ee ty Oa Pe si NOT C 
have Muir’s theorem II. 
4, If further we make all the polynomials in IF, identical, there 
results the relation 
@+at....+ay-Z@+a+....+a, +... 
+ (AP? ZE (a, + a)" + (CD Say 
SRE et on A el Cut à EE yi aac 
à (D) 
When p > n the right-hand side vanishes and we have Muir's theo- 
rem ITI. 
5. US ia ated ate = 1 we have from (D) 

pp (pty REED (pay... CD » + ip 
. 
2! 

= i ha Nr Le) Cr en) HAE (a) 
on Se Œ) 
The value of this series on the left of (E) has been given in deter- 
minant form by Dr. E. D. Roe.! 

1 Annals of Mathematics, Vol. II., No. 6, p. 191. 
