THE INDEX SYMBOL IN THE CALCULUS OF OPEEATIONS. 
15 
So that, finally, 
(H+H.)”={V±V,-2(w-1)}(H±S.)”-‘ 
= {V±Vz-2(7^-l)}{V±V.-2(^^-2)}..{V±VI-2}{V±V,}, 
the upper or the lower sign being taken throughout. And since 
(3±s.)”=(*±y)*(s±|)” 
an equation which might be established by other methods. 
Returning to the expressions for we have 
S”=(V-w+l)..(V-2)(V-l)V, 
S"-‘H,=(V-^^+l)..(V-2)(V-l)V:, 
H”-^a?=(V-w+l)..(V-2XV?-V), 
H«-^a?=(V-?^^-l)..(V?-3V+2)V:, 
S'*-^Et=(V-w+l)..(Vt-6VV?+3V^+8V?-6V), 
H”-'Hf=(V-w+l)..(Vi— 10V?V+12Vi-8V?V+12ViV"+3V'-44V,V-6V?+24V.), 
These expressions are, however, determinants : thus. 

V 1 
W — 
— 
V 1 
W2 
^1 — 
V, 1 
VV 
V: Vi 
V V, 
V 2 . 
E^E,= 
V 2 . 
EEf= 
V 2 . 
W3 
Vi 2 . 
V V 1 
V V 1 
Vi Vi 1 
V Vil 
VV V 
Vi Vi Vi 
V V Vi 
Vi V Vi 
The corresponding formulae for w=4 are easily deduced from these, as follows: — For 
S^Hi, S^Hf, HHi, add to each determinant a row (V, 3, 0, 0), and a column having 
V for its upper constituent, and for the remainder a repetition of the first column of the 
determinant to which it is added. For Hj the additional row is (Vi, 3, 0, 0), and the 
additional column Vi with a repetition of the first column of The formulae then 
are 
V 3 . . 
W3W 

V 3 . . 
>p(9>pt9. 
V 3 . . 
WW3_ 
V 3 . . 

Vi 3 . . 
V V 2 . 
V V 2 . 
V V 2 . 
Vi Vi 2 . 
V Vi 2 . 
V V V 1 
V V V 1 
Vi Vi Vi 1 
V V V. 1 
Vi V Vi 1 
V V VV 
Vi Vi Vi Vi 
V V V Vi 
Vi Vi V Vi 
V Vi V V, 
D 2 
