544 Proceedings of Royal Society of Edinburgh. [ sess . 
we find 
s 0 • A.iA 2 A 3 — « 1 (A 1 A 2 + A 2 A 3 + AgAj) + Sg^ + A^+ A 3 ) — s 3 = 0, 
and three other equations like it, and so, by elimination of A 1 A 2 A 3 , 
A 4 A 2 + A 2 A 3 + A 3 A 4 , X x + A 2 + A 3 , obtain 
5 0 S 1 S 2 S S I 
5 1 S 2 S S S 4 J _ Q 
5 2 S 3 S 4 S 5 
s 3 s 4 S 5 Sq | 
The general identity which is at the bottom of the whole matter is 
thus seen to he 
s a - s a _ l 2X l + s^gSAjAg - .... + ( - )%_ n A 1 A 2 . . . A n = 0, 
where 
= PiK + PzK + . . . + p n x n a . 
3. On looking into this, however, it will be found not only that 
the expression on the left vanishes, hut that it is the sum of n 
expressions each of which vanishes. These latter are all of one 
type, and the proposition to which we are thus led by a further 
step backwards is — 
If the simple symmetric functions of n elements X 1} A 2 , . . . , X n , — 
that is to say , the functions 
1) ^AiA 2 , 2A 4 A 2 A 3 , 
— be multiplied by consecutive descending powers of any one of the 
elements , and the products be taken alternately + and — , the 
aggregate vanishes. 
For, making the chosen element the last, and denoting the sym- 
metric function of the r th degree by C w>r , we have 
^n,r ^■rE'n— l,r— 1 d“ C w _ 4>r , 
and consequently the aggregate in question 
h n a Cn,o “ A n a-1 C n>1 4- A- 2 C n>2 - ...... 
