PARTICULARLY THOSE OP TWO VARIABLES. 
859 
while the corresponding factor on the right-hand side is 
_1 -(X s +X' s +X'VPX"',) 
p s ] e 2 k; 
and these are equal in virtue of the relation 
^+^+^+^=x a +x;+r' i +x' / ;. 
As an example, by the substitutions a?j+log p 1>s for x lt . . . , X 8 -\- —; log p s for 
7 / % 7 i % 
. . . , (235) gives 
7 Tt 
n^> 
v x , ... , v, 41, . . . , V, 
t=r u=r 
+ S Sm> 
t =1 u -1 
t=r u=r 
-fS Sn$ 
f = 1 U = 1 
• ? 
M+Sn®(^ 
A 3 , . . . 5 Aj, , . 
• • 5 A/ + 1, . 
■ • , \\ 
• J 
avy «=i 
v z , . . . , v s 4 1 , . 
. • , Vf , . 
. . , Vyj 
Ao, . . . , Aj. , . 
. . , A*41, . . . 
, A„ +1, . . . 
v 2 , ... , z^+ 1 , . 
. . ,v t , . . . 
> Vn 5 • • • 
, Vyj 
A x 
41, A 3 41, . . . , 
Ajf + 1, A/, A,,, . . 
. , A,. 41\ 
?l 
, , . . . , 
v 3 + l , v h v u , . . 
. ,V.y / 
A 
+ Sn qA +1 * Xs+1, • • • ’ x,+1, 
t= 1 V*'! » "3 
+nd> 
, A,. -|-1 
Z^l » Vq_ 5 • • • J Vs 4 • • • > . . . f Vy 
Aj^-f-1, A 2 + l, . . . , A*+l, . . . , A-+1 
(238). 
, .. . , v 3 + l, . . . ,v r 
: same expression with A, N, X written throughout for X, v, x respectively j 
The same remark with regard to (238) may be ma.de here as at the end of § 8. 
68. We obviously have from the definition 
1 
j 5 " 
A 3 ,.. 
. •, V\ 
)x lf Xfy . . 
• 9 X r 
L\ y i» 
v v • ■ 
> • i V r J 
= T 
m 1 = —00 
where <fi., <tv_ x are functions of the orders r, r —1 respectively, and 
7a 3 , 
a 3 , ., 
.. , AA , , , 
• ■ 
7 . j X 2’ X B> • • • > X r 
• j *Vy 
(239) 
x s =x s + 2vh ~ h log p u 
in r 
(240). 
J 
Putting all the numbers in the characteristic of <E>,. zero, we have 
