of Edinburgh, Session 1882-83. 
237 
1. In order to establish this proposition it will he necessary to 
premise the following lemma : — 
If ttp a^, ttg, . . o be roots of the equation 
+pxf~‘^ . . .+p^, 
then 
K{n>m) +p^ . +^ 3 . /i^_3 + . . . +p^ . , 
where is the sum of the homogeneous products of a^, a^, ag, . . . 
of n dimensions. 
The proof of this may be briefly indicated thus : — We have 
x”^-ppe^-'^ -ppe^-^ -ppT-^ - . . . -jv 
= {X - aj) {x - ttg) {x-a^) ... {x- af) . 
Writing ^ for x and multiplying off by x"^ we get 
1 
1 - ppe - ppS‘ -ppx? - ... - p^p!^ 
1 _J \_ 
1 Oi-j^X 1 (Xpe I “ CLpC 1 (xpXj 
Thus 
(1 + . . . + h^_y^. + . . . + h^_2 , . x^~^ + . xS^'“ + 1 . x^~^ 
+ . 02" + . . .) (1 -ppx -ppe^ - ppS> - . . . -p^ . 02"*) = 1 ; 
hence, since this is an identical equation, equating to zero the coeffi- 
cient of 02 " in the first member, we get 
K -Pl-iu.^-p^. h„ 2 - • ^«-3 - • • • -Pm 0 , 
or, 
hn=Pi-K-l+P2’ + ^^n-3 + . • • +Pm>h^_„^. 
2. We can now establish our proposition — to express x'^(n>m) in 
the manner indicated above. 
For greater clearness we shall first consider a particular case — 
02^ =^px^ -I- qx^ -f ro2 + s . 
