Mr. Boots on the Analysis of Discontinuous Functions. 137 
F(v) = — (— ya o*"v’[b(x)]¢' (x) 
dx. 
x? 
=— (— oe (1 — xeP + = e? i293 a, 3) F (d(x) |p (2) 
fl) 
=r [pay] 4x"[6(0)]6 (2) + 5 Exe [Ooo (4g WOW (OD) 
by virtue of the relation @ (fee = o(n)e”. 
The theorems of Lagrange and Laplace for the expansion of functions are 
particular cases of the above, which, it is seen, includes an infinite diversity of 
particular theorems, corresponding to the different forms which may be attributed 
to ¢(7). To deduce Lagrange’s theorem, we have 
u=«+hf(u). 
“u—hf(u) =z. 
Hence in (11) writing w — hf(w) in the place of f(w), or « — hf(x) in that of 
J («@), we have 
x=ax— $(x) +hAf[G(2)]- 
Let ¢(7) = x, then x =hf(x), and therefore 
R(u) = F(x) Af(a)P'(x4)+ - £ p(aye'(2) dete (12) 
which is Lagrange’s theorem; but every other form of (x), which makes 
(©) — hf [(xr)] real, and therefore, at least, every real form of (x), will give 
a true expansion. 
To deduce Laplace’s theorem, we have 
u=v(r+hfiu)), -. V*(u)—hf(u) = 2. 
Here, therefore, ¥'(2) — hf (x) must be written for f(a), whence 
x=er—v¥'[¢(x)] + h/[¢(2)]. 
Assume (7) = y¥(x), then x = hf [¥(2)], and substituting in (11) 
v(m) = FL) TASH) Mo) (0) +5 So) Vw) Y@).(13) 
which is but one of an infinite number of possible developments. 
VOL. XXI. T 
