138 Mr. Bootr on the Analysis of Discontinuous Functions. 
If, in (10), we make #(~) = 7, we have 
d 
and it may be interesting to observe, that the general theorem may be proved from 
the particular one in the following manner : 
Let w = $(v), then, since f(u) = «, we have f[¢(v)] = wv. Hence, in (14), 
changing w into v, and writing fp for f, and F@ for Fr, we have 
d ee ) 
r[9(v)] = —(— ge) HO" F [He 9'(2) 5 
or, 
d 
Fu = — (- a, \eaeete ean F'o(a)9!(2r). 
da 
In this way the general theorems of this paper might have been deduced from 
the particular theorem of Lagrange, but it would have been difficult to foresee 
such a generalization, until, by other methods, it had been shewn to be possible : 
while the connexion of these results with Fourier’s theorem is in itself a very 
interesting fact. 
4. Asa final illustration of the general theorem (11), let it be supposed that 
we are in possession of the solution of the equation /(w) =, which we shall 
represent by « = v, and that we desire to express the solution of the equation 
fu) +F(u) = # 
in a series which shall converge rapidly when f(v) is small. 
Writing in (11) wu for F(w), and f(w) + f(w) for f(w), we have 
x= 0 —S(¢(2)) —f(9(2)). 
Let « — f(¢(x)) = 0, then 9(a) =v, and x = —f(u); therefore, 
ud ao Wah ah ahh Dy he aie ee ee a 
eC) da hs 1.2 Tat XY) dg 1.23 dat) dv a a2 
It may be worth while to observe, in conclusion, that the general theorem (10) 
may be expressed in the somewhat more convenient form 
Nee eee ‘4 
F(u) = eae ce SoS 5 [o(2)]9'(2), (16) 
as is evident on expansion. 
