414 
Proceedings of the Boyal Society 
is the single symmetric function which we are in the habit of de- 
noting by %a m [Py v . 
In the like case the Determinant on the left and all the Deter- 
minants on the right take the well-known form of simple alternants. 
We thus obtain a new and widely general theorem in regard to 
the latter functions, viz. : — The prodnict of a simple alternant and a 
single symmetric function of its variables is expressible by a sum of 
simple alternants , whose indices are got by arranging the variables in 
every term of the symmetric function in the same order , and adding 
the indices of each term to the indices of the original alternant , the 
first to the first , the second to the second , and so on. 
As an exceedingly simple instance of this we may show how to 
find by means of it the cofactor of 
1 
a 
a? 
1 
a 2 
a 5 
1 
ft 
ft 2 
in 
1 
ft 2 
/2 5 
1 
y 
y 2 
1 
f 
y 5 
Since both divisor and dividend are alternating functions, the 
quotient must be a symmetric function. And looking to the prin- 
cipal terms of divisor and dividend it is evident that /3y 3 , and 
therefore 5/2y 3 , is a part of the quotient. But the product of the 
divisor and this part of the quotient is by the foregoing 
l 
a 2 
a 5 
l 
a 3 
a 4 
a 
a 2 
a 4 
l 
ft 2 
(P 
— 
l 
ft 3 
ft 1 
— 
ft 
ft 2 
ft 4 
l 
r 2 
y5 
l 
r 
y 
y 2 
y 4 
Subtracting this from the dividend and writing the remainder in a 
shorter notation, we have 
| a^Vl + UWI- 
In this the divisor is, as before, seen to be contained fi 2 y 2 times, and 
+ + 
therefore %jp y 2 or | a 0 /3 2 y 2 1 times. Multiplying the divisor by 
this new portion of the quotient we obtain 
| a°/3 3 y 4 | - | al/3 2 y 4 | ; 
so that our new remainder is 
2 1 ct'/jy | . 
