THE INDEX SYMBOL IN THE CALC'ULUS OE OPERATIONS. 
oO 
and consequently 
[A. , 
I 
1 > 
\ -^* + 1 1 
I 
I y-' A„ 'i 
[Pi+,n ' 
\Pl 
P 2 ' 
P2m^ 
' ^i+m+1 
" \Pl 
P2 
The expression ^> 2 ) * •• may be resolved into its partial frac- 
tions in the usual way by writing 
C,: 
Pi 
iPl-P2)iPl-P3) --iPl -P2m) 
and then it takes the form 
Cl 1 C2 , C^m 
, C,= 
"2 
{P2-Px)^P2-P2)-'^P2-P2m) 
cdi — 
Px -P 2 
-P 2 
Pi P 2 
gumg 
C.+C y+.., 
Pi pI ) 
am,o\\pj ^>2 ' / \Pi P 2 " 
Cl 
pTi—i+l ' pn—i+l 
P»+)n + l J" 
^ 3. The case q/ s^ being any f unction of x, y. 
Although it seems doubtful, on account of their complexity, whether the following 
formulae are likely to be of much practical use, it is still worth while to complete the 
theory by considering the most general case where s,, instead of being linear, is any 
function whatever of y. 
Since 
d'^ d d ds d 
^ dx^ dx' ^ dx dx dx 
( 
dx' ^ dx dx dx' 
therefore 
2s,s£,= (s4-|)4+(« 
Again, 
d^ d d’^ dso s. 
d dss^ d’^ 
dx' dx^ dx dx'^ 
d d'^ ds^s d'^ 
dx ' dx^ dx dx"^ 
