THE IJ^DEX S™B0L IN THE CALCHLHS OF OPEEATIONS. 
27 
And if we assume 
%=(^o 
the values of the coefficients, as affected by the operative factor, may be calculated by 
formulae given above. 
§2. 
Let s, Si, . . be any linear functions of the variables x, y, such as 
Si=cCiX-\-^iy, 
and let 
d d 
It is proposed in the first place to investigate expressions for symbols of the form 
d^ 
Sy Sa * * Si 
in terms of the us and vs. 
First, as is easily seen. 
dx'~^ dif 
d'^ d / d \ . , 
therefore 
^^^dxdy—^dx' dy ~ dy—^dy-^^dx~^^^ dx ’ 
‘^^^^dxdy — ^dxi^^^dy “') 
= ?« ( Vi — /3 , ) -f- ('*b — « 1 )• 
Hence the quadratic system. 
'^OO'^Ol ^^2 ^^oov'^^oi *^01 ji 
or 
d^ 
^^10 '^11 — ^^io(^ii ^n}~l~^io(^ii 
^ 20^21 ^2 — ^ 2 o (^21 
yy yy cy 
{a,x^+2h,xy+c,y^)-^-\-2{a,x‘^-[-2h,xy+c,y'^)-y^-{-{a^x‘^-]r‘^b^xy-\-c^y‘^)j^^ 
— 2 ^ 00^01 “ 1 " ^ 10^11 “F ^ 10^^11 “ f “ 2 ^ 202^21 ® q , Wqq 21^20- 
Again, developing the following expression by Euler’s theorem, and remembering 
that, since the .ss are linear, their second and higher differential coefficients vanish, we 
have 
