MR. BOOLE ON A GENERAL METHOD IN ANALYSIS. 
229 
Under the first 2 in the second member the coefficient of f 1 is Tf m {r), and under the 
second 2 the coefficient of is Xf m ~ i(V), wherefore the aggregate coefficient of in 
the expansion of is 
Again, we have 
lO) 
/(T-be)(^+£)=2/4T)f 2 (T+£), 
= 2 f m + 2 f m (ir)f* + 1 
wherein the aggregate coefficient of is 
fm {<7r)h m K -\-f m _ i ( tt) . 
Equating this expression with (3.), we have 
fm {nr)\ m T-\f m _ 1 (t) = Tf n [r) 1 0) 
by (2.), 
(3.) 
or separating the symbols, 
/»(*■) 
fm—lf) fm — lf) . 
— 7T 
(*— i]/-iW 
(X w — 1)tT 
(4.) 
which expresses the law of formation of the coefficients. 
The first term ffnr) is equal to f{f) : this may be proved by induction from the 
particular cases of (•r+g) 2 , (^r+f) 3 , See., but perhaps more rigidly thus. Let k be a 
symbol such that kf{r) =.f(r) . Then the first term of the expansion of 
is kTf(r) ; but by (3.) this term is Tf Q (T)=Tkf(r), therefore 
h7rf(7r)=7rkf(7r) ; 
wherefore t and k are commutative. It is hence evident that k can only operate as a 
constant multiplier, the value of which is independent of the form of f(f). Let 
— then, since f(T-^-^) = T-\-^, it is evident that k— ] , wherefore 
and the expansion is completely determined. 
Cor. If the symbols t and § combine according to the law 
§Af u =ff+ A t) S u, 
At being any constant increment, then 
A A 2 A 3 « 3 
/(’ r +?)=/W+^/W?+^ ! r2/(^) • (*■) 
the interpretation of being 
/(tt + Att)— /(tt) . 
At 
&M- 
For X/(t) — f(T A t) . Hence X m T=T-\-mAT and (4.) gives 
