VARIABLES IN CERTAIN LINEAR DIFFERENTIAL OPERATORS. 
39 
If, as in the earlier part of the last article, -q, { are simultaneous increments of 
X, y, z, we may look upon 
/y» /y* y <y* y 
0^10, «^oi’ '^30’ ‘^11’ "^03’ ■ • • 
as a number of independent quantities ; upon 
and 
i/i05 2/30’ 2/in 2/o3> • • • 
^10, ^01’ ^30> ^11’ %3> • • • 
as determinate functions of these quantities ; upon y, I as three quantities connected 
with one another, and with x.^q, ... by a relation of which (63), (64), and (65) 
are equivalent forms. 
Of the quantities x^q, Xq^, x.iq, ... let one x^^ alone receive an infinitesimal variation : 
also of £. y, let y and ^ be kept constant so that ^ receives a consequent variation. 
Some or all of y^Q, ^/q^, y.^Q, . . . and some or all of ^on 2 -' 3 o> • • • will also receive 
consequent variations. From (63) we thus obtain 
from (64) 
0 = I2/01 + 2/11^ + + 2/31^^ + + 
.}3^ 
+ 
and from (65) 
y I > I ^1/20 y -2 I i 
^ ^ _j_ ^2 
doOy^ ctiX/fg dj.fg 
«+tf+ 
0 = { 2=10 + 2%of + hiV + + ^iin‘ ■ • • ] Sf 
The three relations are identical. Let us study the identity of the first and third 
We obtain from them that 
'in 
dXr 
t I tz _T_ t I 2 I 
dxj^ dxj + .yy +■ ■ ■ 
= {^10 + 2z,q^ + z-^^y + 3zgo^^ + ‘^z.2i^y + z^^y'^ + . 
. . (80) 
for all values of £ and y ; and, consequently, that if by aid of (65) the right hand 
member be like the left, expanded in powers and products of powers of ^ and y, the 
coefficients of corresponding terms on the two sides will be equal. In other words, 
each dzpjdxrs is the coefficient of the corresponding ^Py'i. 
Now, in the equivalence of operators. 
