754 
PROFESSOR BOOLE ON THE DIFFERENTIAL EQUATIONS WHICH 
Now it may be shown that 
( m — 1 \ , T . nut 
7T ) — V sin — . , > 
- L =% sin ( m tt) Y r , 
1 — 2Y cos - + V 2 ' n ' 
the summation with respect to r extending from r = 0 to r = oo . Hence the first mem- 
ber of (V) may be developed in the form 
[n — 1 v m ~ 1 sin ^ ^ V r+1 dv 
f" . ,T T , + , 7 £«> v ™+{r+l){n-l) dv 
J, * V * =(!f J„ (!+»")'+' 
r ^ ?n + (r + l)(w— l) ^ r ^ r— m+1 ^ 
wr(r+l) ’ 
and 
Now 
r r i 
1 v m 'V r+1 dv — x r+ 2 
(m+ 1 + (r + 2)(n— 1)\ 
! r l 
(r — m + l)\ 
{ n ) 
l » / 
Jo 
nT(r+2) 
n(r+l) 
Hence the total coefficient of x r+l in (V) is 
r 
r /«+(r + ])(»-l)\ r /r-m+l\ 
+ (r, + !)(»-!) V n ) \ n ) 
nr(r+ 1) 
(m—r— 1)tt 
,/ >» + (r + l)(n— l) \ r / r— m + l \ 
'4t!) ‘ 
. (m—r — \)ir / m + (r + l)(n— 1) \ p/ r— m-\- 1 \ 
V n / V n ) „ -> 
m + (r+l)(n— 1) 
i(r + l) 
nT(r + 1) 
r + 1 
and therefore that of x r is 
nT(r) ^ r 
,+r(n— 1) \ p/ r ~ m \ r( r ~ m \r( ._ r ~ m \ 
n / \ » / 1 \ n / \ _ n / 
Now 
