233 
STATED MEETING, WEDNESDAY, NOVEMBER 30, 1859. 
James Hentuorn Topp, D. D., President, in the Chair. 
The Rev. Ropert Carmicnatt, F. T.C.D., read the second part of 
a paper— 
ON CERTAIN METHODS IN THE CALCULUS OF FINITE DIFFERENCES. 
Szcr. II.—On the Application of the Caleulus of Finite Differences to 
the Symbolical Reduction of certain Definite Integrals. - 
Tux theorems principally employed for the deductions of the results 
contained in the following section are two, fundamental in their cha- 
racter as regards the Calculus of Finite Differ ences, and easily proved, 
namely— 
F (e?z) .m*™=F (m).m’, (1.) 
and 
F (A). m7 =F (m-1). m*, (II.) 
where F is any algebraic function of the quantity it contains, m any 
constant, and A the ordinary symbol of this Calculus. 
1. By the first theorem, if we were required to determine the value of 
any definite integral of the form 
[2 F @) ade, 
ay 
we see that this integral is instantly reducible to the symbolic shape 
6 F (e?2) dz ; 
a1 
and if the quantity a be supposed to be independent of the limits of the 
integral x, x., transferring, as is legitimate, the symbolic operator out- 
side the sign of integration, we have, as the symbolic result, simply 
Da vot — ao 
Fe.) 
the further evaluation of which will depend upon the particular form of 
the given function F. The result now obtained admits of ready veri- 
fication by the substitution for F (x), F (e+), of their equivalents, de- 
rived from the formula 
F (u) =F (0) +¥F’(0). ¥ +E" (0). 25 $F"). sates 
and its value consists in the circumstance that the question proposed has 
now become reduced to the mechanical working out of the product ofa 
known operation, upon a simple known subject, that operation proceed- 
ing according to a known and practical method. 
B,T. A. PROC.—VOL. VII. 2M 
