CALCULUS OF FINITE DIFFERENCES.

DIRECT METHOD OF DIFFERENCES.

α

nux

(1.) ▲2('U2+2u2+&c.+"u ̧+const.) = ▲ ̧1u„+&c. +▲,"

x

▲a.u2 = a▲ ̧u, if a is either independent of a, or such a function of a, that its value is not changed, when a becomes

A x ( U x • U x + 1 • • • Ux + n) = (Ux+n+1 — U x) U x + 1 • • • U x + n2

-1

2

▲ (ax+α-xn−1+&c. + a ̧x+a) = n(n − 1)a„x” −2 + &c.

▲ (α„x”+α-x2¬1+&c.+a ̧x+a)=1.2.3...n.a„i

from which it appears that the nth difference of an algebraic function of n dimensions is constant, and therefore that the differences of all orders superior to the nth are equal to nothing.

(3) ▲ ̧1u ̧·3u........"u ̧={(1+1A)(1 +°A)...(1+TMA)−1} ‚1u ̧.·3Ù........' in which 1A, &c. are merely used to denote that 'u,, &c. are to be respectively annexed to them after the expansion.

2

m

|

α

mu

▲"1u ̧3u .."u ̧={(1+1A„)(1+°▲„).....(I +”▲ ̧) — 1}".1u ̧·3u..........”u ̧(Tr. L. App. 344-56; L. D. C. 499-507; L. C. D. 882-7.) (4.) Series involving the differences and differential coefficients of a function.

ท

1.2.3..(n+1)

+1

d+1u, + &c.

(Tr. L. App. 357—61; L. C. D. 929—38.)

m m-r,r

AP,'u=S, C ̧‚¿('u ̧·▲ ̧1u ̧) :

this notation will be explained in the Appendix.

An

= { (1 + A„)(1 + A1) — 1} u ̧ ̧.·

▲nu,= {(1 + ▲„)(1+A1)−1}"u‚‚· (L.C.D. 919, 20.)

m

= { (1 + A2,)(1 + A ̧ ̧)....... (1 + ▲xm) — 1 } "U ̧ ̧ï...

...

▲xyUz‚y= 1⁄2) (A2Uz‚y + ▲x2xy+1) + † (AyUxx + AyUx+1)•

(Tr. L. App. 362-6; L. C. D. 919, 20; 33, 4.)

INTERPOLATION OF SERIES.

&c. Un'

(6.) Let u, u19 U25 and V, V1, V29 &c. vn be corresponding known values of u and v, in the equation u= p(v), in which the form of the function is unknown, and let it be required to determine the value of u, corresponding to any proposed value v,, which is either between the limits v, v

or very near to one of them. Let v, be the value of v preceding then if the intervals between the values of v are small, we may assume

Vx9

u ̧= a+a ̧ (v ̧− v ̧) + a1⁄2 (v ̧− v ̧)2 + &c. + a„(v«− v,)”, which equation, when arranged according to powers of v„, u2=b+b1vx+b2v22 + &c. +b2vn,

becomes

1

in which b, b1, &c. remain to be determined. We have the n+1 equations

then

and

V4-V1

b=u— A.v+ B.vv1 — С.vv1v2+ &c.

b1 = A−B(v + v1) + C(vv, + vv, + v1v2) — &c.
b2 = B−C(v + v1 + v2) + &c.

u ̧=u+A (v ̧− v) + B (vx− v)(v2 — v1)

The value of u may likewise be put under the form

(1)