270 
MR. BOOLE ON A GENERAL METHOD IN ANALYSIS. 
conditions that .r=0, y = 0 shall respectively give u~f(x ), and u=f x (y ) ; differenti- 
ating' (76.) and proceeding as before, we find 
u= 
x lx + 1 + ff #)/i(y) -/(°) 
I + app -t- a 2 j -f bxy 
The general theorem (XX.) applied in the two preceding examples is formed on 
the condition that, the generating function u shall involve in its development positive 
powers only of x and y. This condition introduces into the second member a greater 
number of arbitrary functions than would otherwise be necessary. If, for example, 
it were merely required that the indices of y in the development should be positive 
and ascending, the form of the second member of (XX.) would simply be 
'lf(mn)i’ n6 + n6 ' -{- f) + (s') /. . + i (s') /. 
Which of the two assumptions is preferable or necessary must be determined by 
special considerations. 
As an example of the latter form, let us take the very simple equation 
m(m— \ )u mM _ 2 — a 2 n(n— l)u m _ 2M —0. 
We have 
D (D - 1 ) z 26 'u - a 2 D'(B' - 1 y~ e u = F 0 (s') -j- F l (s') s'', 
. dHl _ x> (Pu — W + F) (x)y 
‘ dx 2 dy 1 i' 2 f 
The solution of this equation will put us in possession of the complete value of u, 
the functions F 0 (F), F x (a?) admitting of either positive or negative indices in their de- 
velopments. If we assume those functions to vanish, we get 
u=<p(y-\- ax) + ^ (y — ax ) , 
which is a particular value of the generating function. 
Many other developments and applications might be here given, were the subject 
of sufficient importance to justify further detail. 
E. Application of the Theory of Series to the Evaluation of certain Definite Integrals . 
Ex. 1. The function (l— 2v cos <y-J-»' 2 )~ re being expanded in a series of the form 
A 0 -}-2(A 1 cos m-\-A 2 cos 2»-j-A 3 cos 3<y-(- &c.), 
it is required to determine the general coefficient A,.. 
We have 
(1— 2j/cos&>+i' 2 )“ w =(l -»6 * v - 1 )-*X(1 __ ^-w- 1 )— 
{ , | / . . n{n+\) . . n{n+\)(n+2) „ „ . , , 0 \ 
= 1 -\-nvz us _1 -j- — y~ 2 — 1 -j- 12 3 V 3 a' w_1 -j-&c .j 
/. - / i , n(n+ 1) 0 o / i , n(n+l)(n + 2) 
X (l + ~ W ~ 1 + - \ # g ■ y2g ~ W ~ 1 + - - _ Es 
and the quantity sought, A,., will be the common coefficient of a and in 
23 -j ) 3 g- 3 W-i_j_g cc> \ 
