of Edinburgh, Session 1881-82. 413 
as there has been in its formation an even or an odd number of 
+ + 
interchanges in the rows of | xgy 2 z z | . The proof is on the same 
lines as that of the former theorem. 
7. The product of a Permanent and a Determinant both of the 
n th order is expressible as an aggregate of n ! Determinants of the 
same order. Thus — 
+ 
+ 
✓y* sy> /y» 
1^2 t*/g 
0 
Vi V2 y s 
C 1 C 2 C 3 
h *2 % 
+ 
a 1 x 1 
u 2 x 2 
a 1 x 1 
acpc 2 
a 3 X 3 
\y 2 
\y s 
— 
b 2 z 2 
b sh 
c ih 
C 2 Z 2 
C 3 Z 3 
VJi 
^2 
C ??J 3 
“iVi 
a 2V2 
a ^3 
a iVi 
^2 
a 3 V 3 
b 1 x 1 
b 2 X 2 
^ 3 X 3 
+ 
hh 
h Z 2 
hh 
c ih 
C 2 Z 2 
C 3 Z 3 
cpx x 
C 2 X 2 
C ‘ 3 X 3 
a x z x 
a 2 Z 2 
a 3 Z 3 
a l z l 
a 2 z 2 
° 3 Z 3 
b x x x 
1j 2 X 2 
1^3X3 
— 
hVi 
\V2 
hv% 
hVi 
C ^2 
C ?J /3 
C 2 X 2 
C 3 X 3 
Here the Determinants on the right are formed exactly like the 
Permanents in §§ 5, 6, and the signs preceding them are determined 
exactly as in § 6. 
8. The last of these three theorems is, so far as one can at present 
see, the most important. It leads, for example, to the following 
valuable application. 
When the elements in the first row of the Permanent are all 
powers of one variable, the elements in the second row the like 
powers of another variable, and so on, the Permanent takes the form 
of a single symmetric function of the said variables, the degree of 
the function being given by the sum of the indices of the powers in 
question. Thus — 
^ fin p v 
y m y n y v 
