480 ON THE SUMMATION OF SERIES, 
i f (2) dees. f (x+h)— =, 3 (h’) {fF (a) ~— 
Jy y 1 
This theorem, involving an arbitrary quantity 
h, which may be made to fulfil any condition at 
pleasure, expresses the definite integral of f (a) 
dx, between the limits of v=w# and w=y. 
Cor. 2. Take h=0 «: 4 = 1; and the’co-efti- 
cients of the even derived functions of f(a) will 
vanish ; this will be the case if h’ be taken equal 
to nothing, consequently we shall have 
z x—l n 
|. P(e) de = sof (a) = s (= 1) Boao 
JY y 1 1.2.3...2n2—2 
( 2n-—3 2n—3 
Ee sa) =f) biQy)) 
eo i (2)- A(f(w) -f(y))-Blf (@) -£ (y)) 
y 
~D(f (@)—f (y) )— By Be. weeesesseeee (15) 
Scholiam. After having obtained the theorem 
in equation (15), I discovered, in looking over 
