42 
OX THE INDEX SYMBOL IN THE CALCEXES OF OPEEATIONS. 
Now the coefficients of the xs are all comprised in the product 
^21 
6422 • • 1 
«!! ^11 • • 
(^21 
i 
(^22 • • 
*^I2 (^12 • • 
• 
• • • 1 
and the terms correspond line for line and column for column. If, therefore, 
-\-{a,x-^h,x,-\-.. )y, 
+ ... 
we may write 
A. A 
cfej dx,^ 
and by a tolerably obvious extension of the symbolical notation used in other parts of 
this paper, we may Avrite this expression thus : 
I ) 2 D , = V2Vi“( 
®21 
/^22 • • 
C4ii / 3 jj . . 
[^21 
(^22 • • 
^12 f ^ l 2 • • 
. . . . 
And generally 
V2V. — V2V.. 
D„D„_,..D,Vr 
Vm V«— 1 • • Vi 
Vn Vm-i • • Vi 
Vm Vn-I • • Vi 
Lastly, the formulse for ss , . . j— in the case of two variables may be directlv ex- 
dx'"-~f‘dyP 
d ” 
tended to the corresponding case of ss .^ . . ^ - — with many variables. 
uJO ^ (XJu ^ • • 
