230 
MR. BOOLE ON A GENERAL METHOD IN ANALYSIS. 
fm(*) = 
fm — 1 H~ Att) fm — i(^) 
toAt 
J_ A/ / X 
“m 
whence the theorem is manifest. 
If A tt= — 1, we find 
/(**+£) =/oO) +/i( 7r )?+TT 2 / 2 ( ,r )? 2 + &c -> 
where fo{v)=f(r), and in general/ m (<?r)=/^_i(^)— 1). 
If A^r vanishes the symbols t and are commutative, becomes and (I.) is re- 
duced to Taylors theorem. 
r <l 
Prop. 2. If g=<p(x)& dx then x and § combine according to the law 
gf(x)u=f(x+r)§u. 
For writing u x in the place of u, we have 
d 
r — 
gf(x)u x =<p(x ) s dx f(x)u x , 
= <p(x)f(x+r)u x+r , 
d 
Y 
=f(x+r)p(x)s dx u x , 
=j\x+r)§u. 
Prop. 3. If 7C— 
T 
n<p(x) s dx —x 
and g=<p(x)i r dx, then n and § combine according to the 
law 
w 
§ f(T)u=f(7T-l)gU, 
e have Now £ combines with x according to the law 
gf(x)u=f(x+r)gu, (5.) 
and § combines with § as if it were a mere symbol of quantity ; hence 
g/(~~~)^ =/( ” g ~ + r) ) s u h y Pro P’ 2 > 
=A*—i)gu- 
This result may also be proved by expanding in ascending powers of § by 
Prop. 1, and operating with on each term of the series. 
d 
d 
ax — ^ # 
Prop. 4. If %— and g—xs dx , then -r and g> satisfy the following relations, 
