THE INDEX SYMBOL IN THE CALCHLHS OF OPEEATIONS. 
37 
1.2 .. (m+l)ssi .. s„ 
dm+l 
dx”^+^ 
d t c d^ -I r, ^^1 *2 • • *»« d 
dx'^ 
, ^ 1 o d^ 1 o dsSi^»»s„i d^ 
+ s , 1 . 2 . . m s §2 . . ^ — 1 . 2 . . m 5 . 
dx 
+ ... 
dx dx^ 
d 
^ dx 
ds \ 
'^Tx) 
1.2. 
. 7?^SlS2 . 
dr 
■ ^'^dx'^ 
d 
^^dx~ 
)1.2. 
. 771SS2 . 
dr 
' dx"^ 
+ ... 
_/«? ds\ / d , fi? 
which proves it in the general case. 
Again, 
ss 
^ dx dy' 
^ dx dy ^ dx dy 
d’^ d d ds^ d 
dx dy 
dsy d 
^ dy^^ dx ^ dy dx 
d d ds d 
— ^ilZ ^ 17 . — ^ilZIZ 
d d 
dy 
d d 
Hence 
^ dx ^ dy 
d d 
~^^dy^ dx~ 'f- 
dx dy 
ds d 
dy dx 
^dxdy \ dx dx) * ' \ ^ dx dx ) 
'dy 
Again, 
ss 
Hence 
/ d ds\ d / d ds{\ d 
\ dy dy ) dx~^ \ ' dy dy dx 
d^ d c?^ dS] d'^ ds^ d!^ 
dx^dy ^ dij^^^^dx"^ dy^^^ dx^ dy ^^^dx^ 
ds d^ d d^ ds^ d^ 
dy^^^^ dx'^'^^ dy^^^ dx^ dy^^^dx'^ 
ds d^ <fej d'^ d d^ 
dy^^'^'^dx^ dy^^^dx'^'^^dy^^^dx‘^ 
goo o — ( s— 2 — ^ 2 s s “ 
D«Si Sa ^^2 dy—ydy ^ dy ) ^^2 
MDCCCLX. 
G 
