Σ

=

α

b(a+1-1)

at

=const.

b(a"-1-1) u2 · U2 - 1 • • • U z + n-2

(Tr. L. App. 373-5; L. C. D. 955—60.)

(23.) Σ.(...)=u. Σου -Σ (Δ.Κ. Σ.υ.+1),

.

=u Συ- Δ. Συ+1+1 " Συ+- &c.

EQUATIONS OF DIFFERENCES.

(25.) The complete integral of an equation of differences of the nth order

0 = p(x, u2, u2+1 &c. U x + n)

must contain n independent arbitrary constants..

(26.) The general equation of the first order and degree is ux+1=a ̧u2+bx•

Let u ̧=v ̧a ̧·α........a ̧-1=v ̧·Pm(am), *

V1)Ĺ„(a‚„)=b ̧

then (Vx+1—Vx) Pm (am)=b ̧, or ▲ ̧v ̧·Þ„(am)=b ̧;

(Tr. L. App. 379-81; L. C. D. 1038.)

(27.) The general equation of the nth order and first degree is

Uz • − U z + n + 1 α ̧· Ux + n − 1 + 2α z • Uz + n −2+ &c. + "α ̧· Ux=b2 ; 2ax Ux ˆax• the integration of which is reducible to that of the equation

..

n

(1)

(2)

Uz + n + 1αz • Uz + n − 1 + 2 α z · U x + n − 2 + &c. + "a2.u ̧=0: which latter equation is always integrable if n-1 particular integrals can be obtained; and if m particular integrals can be obtained, it may be reduced to an equation of the (n—m)th order. The general method may, to avoid complexity, be illustrated by its application to the equation

U z + 3 + 1 α ̧· U x + 2 + εα ̧•? + 3a ̧•u2=b2·

x+1

Let this equation be obtained by elimination from the equations

in which 'C, C, and 3C are arbitrary constants.

From these equations we may obtain by elimination

ux=1C.1Ux+2C.2U2+3C.3U„+ Vx,

If b=0, then V=0, and the equation

u2=1C.1Ụ2+2C.2 U2+3C.3Uμ

is the complete integral of

by

Pm (– 3v,
Pm(-3vm) 2 í

If this equation is known, then 1v,, v, &c. may be determined, for from (6)

these quantities being substituted in V1, and the result added to the complete integral of (2), the complete integral of (1) will be obtained,

Ꮖ

If n-1 of the quantities U are known, the nth may be obtained from the equations (3).

(28.) If the coefficients of (2) are constant, that equation will be satisfied by assuming u=e; then, dividing by e^, we have

2

e" +1a.en-1+ 2a.en - 2 + &c. +"-1a.e+"a=0.

Let e, eg, &c. e, be the roots of this equation, then

u1C.e+C.es + &c. + "C.e

is the complete integral required.

If any of the quantities e,, e, &c. are equal, or impossible, the changes which take place in the value of u are analogous to those in (Int. Calc. 43.).

(29.) If we wish to extend the integral of (2) to (1), then

u2=1C.e† + &c. + "C.e„ +

b

na

1 1 + 1a + 2a + &c. + " (Tr. L. App. 382-5; L. C. D. 1036–52.)

(30.) An equation of the second order and first degree