278 
MR. BOOLE ON A GENERAL METHOD IN ANALYSIS. 
We have here two distinct series. Either of these is an integral of the equation 
(99.), but I apprehend that the second series only can be regarded as an integral of 
the original equation. The first series appears to have reference only to the irrelevant 
factor x. As I am not aware that this distinction has been before observed, and as 
it appears to affect the validity of the canon which assigns to the integral of an 
equation of differences just so many arbitrary constants as there are units in the 
index of its order, I shall make no apology for adducing another illustration. 
Let us take the equation 
■«■*».= 0 (101.) 
We find as the symbolical form, 
& M— 1 
m 
( T +fK=°; 
whence u = 2a ,p m with the relation mn -\-a n . = 0, or a - 
x mb m 1 m— i i m 
m being 0, whence 
_rY, x . *(*-1) x(w—\)(ac—2) , 0 _ \ 
”.- c t l -T +_ T2 K23 + &c j’ 
the lowest value of 
=cd-ir, 
=co*. 
This is obviously a true integral of (101.) for all values of a 1 from 0 to oc ; for when 
a =0 it gives u x — C, and when x is any positive quantity, u x = 0, precisely as it ought 
to do. Moreover it involves an arbitrary constant C; but (101.) being of the 0th 
order, there should, according to the received canon, be no arbitrary constant in its 
integral. 
For the solution of the equation x(x — l'u x =0, we similarly find 
u x =a 0 -\-a 1 x+a 2 x(x—\)-\-a i x{x— l)(.r— 2)+&c., 
wherein a 0 , a l are arbitrary constants, and in general 
2{m— l)a m _ x + a m _ 2 
a ■=. — > 
'« m[m — I) 
this gives 
, 2 a, + a n , , . . 3a. — 2a 0 , „ 
u x = a o + a \X 2 ~^x(x— 1 ) + g — x(x— l)(x - 2) — &c. 
For the values d’=0, x= 1, the above series is reduced to an arbitrary constant ; 
for all other positive values of x it vanishes. Here then we obtain for an equation 
of the 0th degree, a solution involving two arbitrary constants, and derived from the 
two irrelevant factors x and x— 1. 
It would be interesting to inquire whether an equation of differences admits of 
integration by series when f 0 (m) has equal or imaginary factors. After paying some 
attention to this question, I am disposed to think that such cases are not compre- 
hended in any theory analogous to that which I have given for the corresponding 
class of differential equations. 
On the subject of this section the reader may consult a paper by Mr. Bronwin in 
the Cambridge Mathematical Journal, vol. iii. 
