468 
THE REV. B. BRONWIN ON THE SOLUTION OF 
will reduce to 
/M(»' + (n+l)^)«’+#,y)’^ ‘*w=p|”*| X, &c. 
respectively, the reductions being made by the theorem {b.). 
But these equations and every step of their reduction may be derived immediately 
from (1.), (3.), &c., and their reductions by the conversion of symbols before men- 
tioned, which is not a little remarkable. Hence we may derive the solutions of the 
one series from those of the other merely by the interchange of symbols. 
In the equations of which we are now treating, 9(D) and A(D) must be rational 
functions of D ; and we must also have a rational function. Therefore 
we must have 
and 
_l %'(D) 
kx{D)’ 
9(D) = - 
^X(D) 
X'(D) 
In the values of u we have operating factors of the form 
/ XfT)'! -l-rZ-N -1 fM'0)+rh_^ 
(^'4-rA)-' = (9(D)^-)-A(D)+rA-)-*=(x-h^^^^j ^(D)-^=2‘'"^^"VV-'~^‘®9(D)- 
by a well-known theorem due to Mr. Boole. Now in order that these may be prac- 
ticable in finite terms, or as it is usual to say, capable of interpretation, we must have 
zJ <p(D) =q>(D)s’”’^, 
0(D) being a rational function of D, and m being any constant, positive or negative, 
or nothing. This is the most general form possible, and it gives 
A(D) _ mO(D)-f(h'(D) 
^)- “0(D) ’ 
and 
^(D)=|^(m<t-(D)+<l.’(D)). 
f dD 
The expression s*' ^ has been rendered practicable in making r'"* so. 
The only other operating factor which we have to consider is 
h being some constant. Now putting for its value before found, this will reduce to 
X(D) + 4-fjox(D)'l 
x-\ 
m 
a(D) , h 
7 
«(D)-. 
The exponentials depending on and have been considered, and it only re- 
