MR. BOOLE ON A GENERAL METHOD IN ANALYSIS. 
231 
f{*) f^u— -{-m)u, (II.) 
fWg m . . (III.) 
the subject u becoming- unity in the second of the above equations. 
d_ 
X£ dx x d 
For in the last Prop, let <p(x)=x, and n—\, then v— §=x& r Tx, and 
eA*)u=J{*— l)gu. 
By induction, g m f(7r)u=f(‘7r—m)g m u; 
f(7r)g m u—g m f(r-\-m)u, 
which is the first of the proposed relations. Now m being a constant is commutative 
with 7T, wherefore expanding /(*-{- m) in the second member by Taylor’s theorem, in 
ascending powers of %, we have 
/( v) fu ~ f { f(m ) u +/' (m) mi +/" + &c. } . 
For u write u x , then 
r— 
XS dx — x 
= U x> 
XU %-\-y ~ XUjq 
which vanishes if u x — 1. In like manner n 2 u x , n^u x , &c. vanish under similar circum- 
stances, wherefore 
/w? m (i)=r/w(i) ) 
which is the second of the relations in question. 
Prop. 5. The same values being attributed to n and §, we shall have 
/ a \ n 
1)..(V— n + 1 )u=x{x+r)..(x+{n - 1 )r) ^ ) u, 
wherein A x—r. 
We have 
p — X 
ft— > 
(IV.) 
l-p-'x 
£ Vm= — 
Now 
( r — \ -1 d d 
g~ l XU= ( XS dx ) XU = S dx X~ l XU = Z diU, 
therefore 
— r— 
| — e dx 
g-l 7 TU = ; U, 
Now 
r n 
| l_ g'^dx] 
/ 1 \n J A e v 
(r r ) u = y — r — j u - 
(f -1 ’ r) 2 u=g- l ‘7Tg- l ‘7ru=g~ 2 (‘7r — 1 )nu=§~ 2 n(n— 1 )u, 
2 H 
MDCCCXLIV. 
