PARTICULARLY THOSE OP TWO VARIABLES. 
847 
V. V 
should be negative for all real values of the m ’s ; beyond this restriction the —^—' 
quantities p are of any form or value whatever. 
60. From the definition it at once follows that 
J^Ij ^23 • 1 
, . , \ + 2 , . . 
• , *rl 
II 
e 
r 
• 3 Xs> • ‘ 
• , Al 
1 V V v 2> • • 
. , v s , 
• 3 V r \ 
i v \> v v • • 
• 3 V S , . . 
• 3 V,j 
=(—1)M> 
V l> V 2> 
( 221 ) 
the variables being the same throughout. Hence it is obvious that the number of 
distinct functions is 2 2r =4 r . Also 
<S> 
f Ai, ^2> • • 
.,x\ 
IW • ■ 
■ 3 V r J 
■ X n 
■X, 
An 
• • 
• 3 \\ 
fl?l, fl? 3 , . . 
. ,X r \ 
\ v i> 
v z , . . , 
• 3 Vr) 
\ 
. ( 222 ) 
a formula which enables us to distinguish between even and uneven functions. For 
each of the expressions, as X s v s , zero value may arise in three ways, viz.: X*.= 0, v s =0; 
X s =l, v s =0; X s =0, v s — 1; and a value unity arises in one way, \ s — 1, v s = 1; and an 
uneven function will occur when the number of units in the index is odd. Thus if 
P, Q denote the numbers of even and uneven functions respectively 
and therefore 
so that 
p = y + !^=jl yH8 + "-^7 2 - r - 3 3 ,-4 + 
1.2 
1.2.3.4 
Q =n3 - 1 + ^^ 3 - + . . . 
Pq-Q=(3+iy = 2 2r 
P—Q=(3 — l) r =2 r 
P=2 2r-1 + 2 r_1 
Q = 2 2r-L — 2 r ~ l . 
5 Q 
MDCCCLXXXII. 
