238 
Proceedings of the Royal Society 
Multiplying both sides of this equation by writing for its 
equivalent value, and arranging, we get 
=■ ijf + (f)x^ + {yyq + r)P + {jgv + s)x ; 
but by the lemma 
and in particular = h^—phj^-q, = ^ 9/^2 + + r, &c., where 
/q is the sum of the homogeneous products of the roots of 
=px^ + qx^^ + TX + S oi n dimensions. Thus 
= 11^^ x^ + (5/q + r)x‘^ + {rh-^ + s)x + . 
Kepeating the above process on this equation we shall find, on 
again applying the lemma, that 
x^ = /q . x^ + (g/q + rl\ + s)x^ f (r/q + slif)x + sh^ . 
ITow assume 
= hn-Z ‘ P" + • P 
+ (^^q-4 + s^q- 5 )^ + ; 
operating on this equation in like manner we shall get, since by the 
lemma + qh^_^ + rhy,. ^ + = /q_ 2 , 
+ {qhn-z + 
+ {P^n-Z + sK.^X + s/?^_3 , 
which proves the preposition in a particular case. 
The general case is established in a similar way, 
x^ ==ppy"^~^ -{-pp(g^~^ + . . .+p^. 
Multiplying both sides of this equation by x^ writing for x'^ its 
equivalent value, and arranging the terms, we shall get, since by 
the lemma pf-^ +P 2 ~ ? 
+ ' = /q . 4- {pf ^ + p^)x^~^ + OVb -\-p^)x^-^ 
+ • . • + (Pm-l . +Pm) • « +ib. . /q • 
Applying the same process to this equation, it may be shown, since 
iV' 2 +i’ 2 ?il + P3 = ^'3> 
»'"+® = 7*3 .x^-' + {pjl^ +X^3\ +Pi)x^~^ + (pJh +Pjh 
+ ■ ■ ■+(Pm-l- ^h+Pr.- h^^+P„- 7*2. 
