A simpler proof of the above theorem (which may be more shortly written 4 (t) = ☀ (D) eo.t) is obtained by regarding it as a particular case of Herschel's theorem, viz.

$ (e13) = $ (1) + $ (E) 0 . t + $ (E) 02 .

or, symbolically written, (e)

=

0.t

(E) e. The truth of the last theorem is at once rendered evident by assuming Aet to be any term in the expansion of (e) in powers of e. Then since Aet A,E" e the identity of the two series is evident.

But

0.1

$ (t) = 4 (log e1) = ¢ (log E) €o.t

(by Herschel's theorem)

= $ (D) €o.t,

which is the secondary form of Maclaurin's theorem.

As a particular illustration suppose (t) = (e* − 1)", then by means of either of the above theorems we easily deduce

But A"0" is equal to 0 if m is less than n and to 1.2.3...n

Hence

(e'-1)"="+:

1.2...(n+1) 1.2...(n+2)

d

dx

.t”+2+&c......(20).

Hence therefore since A′′u = (ea — 1)′′u we have

Δ"0"+2 d+2u du" 1.2...(n+1)` dx2+11.2...(n+2) dɔ2+2

Anu= +

the theorem sought.

The reasoning employed in the above investigation proceeds upon the assumption that n is a positive integer. The

* Since both A and D performed on a constant produce as result zero, it is obvious that (D) C'= ¢ (0) C=4 (A) C, and 4 (E) C = p (1) C. It is of course assumed throughout that the coefficients in & are constants.

very important case in which n=-1 will be considered in another chapter of this work.

in terms of the successive differences

and the right-hand member must now be developed in ascending powers of A.

In the particular case of n = 1, we have

11. It would be easy, but it is needless, to multiply these general theorems, some of those above given being valuable rather as an illustration of principles than for their intrinsic importance. We shall, however, subjoin two general theorems, of which (21) and (23) are particular cases, as they serve to shew how striking is the analogy between the parts played by factorials in the Calculus of Differences and powers in the Differential Calculus.

By Differential Calculus we have

Perform (A) on both sides (A having reference to t alone), and subsequently put t = 0. This gives

$ (0) + $ (A) 0. du, + $ (A) 02 d3u«

$(A) ux=ux • $ (0) + ☀ (A) 0.

dx

of which (21) is a particular case.

+

1.2 dx2

+ &c...(24),

on each side, and subsequently put t = 0;

d

dx

12. We have seen in Art. 9 that the symbols A, E and

or D have, with certain restrictions, the same laws of combination as constants. It is easy to see that, in general, these laws will hold good when they combine with other symbols of operation provided that these latter also obey the above-mentioned laws. By these means the Calculus of Finite Differences may be made to render considerable assistance to the Infinitesimal Calculus, especially in the evaluation of Definite Integrals. We subjoin two examples of this; further applications of this method may be seen in a Mémoire by Cauchy (Journal Polytechnique, Vol. xvII.).

Ex. 6. To shew that B (m +1, n) = (− 1)" Am1, where m

n

∞

Ex. 7. Evaluate u =

0

29+1

Am dz, m being a positive

2 z" + n2

integer greater than a; A relating to n alone.

Let 2x be the even integer next greater than a +1, then

Now the first member of the right-hand side of (26) is a rational integral function of n of an order lower than m. It therefore vanishes when the operation A" is performed on it. We have therefore

This example illustrates strikingly the nature and limits of the commutability of order of the operations

and A.

Had we changed the order (as in (27)) without previously preparing the quantity under the sign of integration, we should have had

which is infinite if a be positive.

The explanation of this singularity is as follows:

If we write for Am its equivalent (E-1) and expand the latter, we see that

Amp (a, n) da expresses the integral

of a quantity of m +1 terms of the form A, (x, n+p), while

Δη

0

P

Amp(x, n) da expresses the sum of m+1 separate integrals, each having under the integral sign one of the terms of the above quantity. Where each term separately integrated gives a finite result, it is of course indifferent which form is used, but where, as in the case before us, two or more would give infinity as result the second form cannot be used.

13. Ex. 8. To shew that

$ (E) 0′′ = E$' (E) 0′′−1.

.(28).

Let A,E' 0" and ErA,E10" be corresponding terms of the two expansions in (28). Then, since each of them equals Ar", the identity of the two series is manifest.

Since E=1+A the theorem may also be written $ (A) 0′′ = Ep' (A) 0′′−1,

and under this form it affords the simplest mode of calculating the successive values of Am0". Putting & (A) = A”, we have

and the differences of O" can be at once calculated from those of On-1.

Other theorems about the properties of the remarkable set of numbers of the form Am0" will be found in the accompanying exercises. Those desirous of further information on the subject may consult the papers of Mr J. Blissard and M. Worontzof in the Quarterly Journal of Mathematics, Vols. VIII. and IX.

EXERCISES.

́ ́ 1. Find the first differences of the following functions;