352 
PROFESSOR A. R, FORSYTH ON 
1, 2, , 2p+1. Evidently al r ^=al^ r , and there are therefore p(2p+l) functions 
al, %s ‘, these, together with the 2p +1 functions al„, are the functions of the theory. 
(They are, of course, not all independent; the complete system of relations among 
them may be found in the fifth section of the first chapter of the memoir already 
quoted.) 
Further 
L halr _+v p 1 bx K 
al r bu s 2 A=i«r— x K bu s 
and 
in which 
-v/b(%) W 1 
a=i ++\) (x\—a s )¥'(a s ) (a r —x^) 
l s al s z al rs 
P '(a s ) al r al s 
bcd r _ l s y 
. HQ , , V di S Gl r S .... 
ou s r ( a s ) 
s may have any value 1, 2, . . . , p 
r >> >> E 2, . . . , 2p+l, 
(19) 
but r, s may not be equal. If r<p, this serves as a verification of (14). 
Again, since x v x 2 , . . . , x p are the roots of (11), 
s=p i a i 2 
^ f'S ul/ S 
1 = 
(x — X^X—Xz) . . . (x—Xp)(f)(x) 
s=i (x—ci s )Y(a s ) (x—cc^)(x—a n ) . . . (x — a p ) ?(«) 
In this write x=ci p+r , (r= 1, 2, . . . , p+l); then 
al\ 
P+r- 
_ ( Pi^p+r) 
s=p 
= l—X 
l s al s " 
= i (a s —ap + r)V'(a s ) 
( 20 ) 
which expresses p -\-1 functions ctl p+r each in terms of the p functions al x , al z , . . . , ctl p . 
By (20) and (14) we have 
ahp+r — 1 + h 
=p 1 tu 
=i a s a p _i_ r bu s 
( 21 ). 
[17. A simpler form can be given to this equation by the introduction of a series ol 
P + 1 new variables provisionally given by 
