THE INDEX SYMBOL IN THE CALCHLUS OF OPEEATIONS. 
25 
it follows that, if 
the coefficients of u will be given by the equation 
Still more generally, 
F(V, Vi)?i=2|F^w, (w— . 
So that the eifect of the operation 
(a5..IEH,r, 
or, as it may also be written, 
± ± 
will be exhibited by replacing V, Vi by w, and respectively in 
either of the values given for [ab . . 3(E E,)'" in pages 16 or 19. 
In order, therefore, to solve the differential equation 
{ah .. XH El)'”^t=0, 
we must first reduce it to the form 
F(V V.>=0; 
then sohing the symbolical equation 
K-'F(VV:)=0 
(where K is the coefficient of the highest poAver of Vi) with respect to Vi , and calling 
the roots /(V), / 2 (V), or simply/, Ave have 
u— 
AV 
►'here 
C,= 
F(V; Vi)~K(V,-/J(Vi-/ 2) .. (Vl-/») 
“ K w -/i+ •• vw»r’ 
/T-' n _ n 
\J<) — 
(/) fm) 
r^ni— 1 
2 
(/a Ai)(/2 im 
In order to evaluate the expression for ti, let ^,,^ 2 , be values of V which make 
/, /j, ../„ vanish. They may in fact be called roots of / ,/, . ./I ; but as these func- 
tions are generally irrational, they cannot be replaced by the products of factors of the 
form SJ —p. In general there Avill be only one quantity p for each function /; because 
if / be rationalized, it Avill give rise to a function of the degree m ; but although the 
equation so formed aaIII in general have m roots, (m— 1) of them aaIII in fact be extra- 
neous to the particular equation rationalized, and belong one apiece to each of the 
remaining equations of the system. 
