462 
THE REV. B. BRONWIN ON THE SOLUTION OF 
second, and the result is 
and so on. Therefore if f{^) be any function developable in integer powers of zc, 
positive or negative, we shall have 
f{zz -\-'k)uz=. 7~^f{zs') T^‘u, 
or 
7^f{zS -\-y)UZ=f{7Jo)T^U («.) 
/ — 1 
'P gives and we may regard either r or (p as the quantity 
given from which the other is to be found. 
A'pplication of the preceding Theorem to the Solution or Reduction of an extensive class 
of Linear Differential Equations. 
To abridge we shall put 
_-o{a-\-nk\ 
\ 7 u ~l~u) a-\-li^ . ..{ts -\-a-\-nh'^ — P 
{zs-\-a) ^{'!!S-\-a-\-h) * (zs-{-a-{-nh) ^ = P 
V 
_ji{a->rnk\-^ 
] 
and similarly in other cases. But it is to be observed that we may resolve the last 
into 
A Ai , A„ 
-CT + a ' zT-^-a + k 
•ST + a + nk 
which in practice may be more convenient. The operations implied by the reciprocal 
factors may be readily performed by a well-known theorem due to Mr. Boole. 
Thus 
Now let 
f{-m)'!SU+pf,{z!s){m-\-nk)T^U=li, ( 1 .) 
where f{-^) and /X^) are any rational functions of sr, and k may be either positive or 
negative. To reduce this, assume 
u=(rs-{-k){ijj-\-2k) {7n-\-nk')v. 
Then by (a.) 
{zs -\-k) . . . .{zu -\-nk)v=.zs{xn -\-k) {m-\-{n — \)k)T^v. 
« 
Substituting these values in (1.), and operating on both members of the result with 
we find 
f{-^)v+pf{z.yv=v\^^-^ X 
( 2 .) 
