first order is made to depend upon that of a linear equation of the second order whose second member is 0 by assuming

And it is remarked that the two constants which appear in the value of v, effectively produce only one in that of u. Prove this.

13. The equation

Ux+2 − (a¤+1 + a ̄*) Ux+1+Ux= 0

may be resolved into two equations of differences of the first order.

14. Given that a particular solution of the equation

x+1

Ux+2 a (a* + 1) Ux+1 +a ux=0 is u= ca

deduce the general solution.

x(x-1)

2

15. The above equation may be solved without the previous knowledge of a particular integral.

16. The equation

U x U x + 1 U x + 2

= α ( u x + U x + 1 + Ux+2)

may be integrated by assuming u = √√/a tan v ̧•

17. Shew also that the general integral of the above equation is included in that of the equation u+3-u0, and hence deduce the former.

18. Shew how to integrate the equation

U x + 1 U x + 2 + U x + q2 U x + U x U x + 1

=

m2.

CHAPTER VIII.

OF EQUATIONS OF DIFFERENCES OF THE FIRST ORDER, BUT NOT OF THE FIRST DEGREE.

1. THE theory of equations of differences which are of a degree higher than the first differs much from that of the corresponding class of differential equations, but it throws upon the latter so remarkable a light that for this end alone. it would be deserving of attentive study. We shall endeavour to keep the connexion of the two subjects in view throughout this chapter.

Expressing an equation of differences of the first order and nth degree in the form

n-2

(Au)" + P1(Au)" ̄1 + P2(▲u)" ̄2 ... + P2u = Q ......... (1), PP P and Q being functions of the variables x and u, and then by algebraic solution reducing it to the form

... 2

(Au —p1) (Au —p2) ... (▲u - pn) = 0......... (2), it is evident that the complete primitive of any one of the component equations,

--

= 0

Au-p1 = 0, Au-p2 = 0... Au — Pn · .... (3), will be a complete primitive of the given equation (1), i. e. a solution involving an arbitrary constant. And thus far there is complete analogy with differential equations (Diff. Equations, Chap. VII. Art. 1). But here a first point of difference arises. The complete primitives of a differential equation of the first order, obtained by resolution of the equation with dy respect to and solution of the component equations, may without loss of generality be replaced by a single complete primitive. (16. Art. 3). Referring to the demonstration of

dx

this, the reader will see that it depends mainly upon the fact that the differential coefficient with respect to x of any function of V, V,... V, variables supposed dependent on x, will be linear with respect to the differential coefficients of these dependent variables (Ib. (16) (17)). But this property does not d remain if the operation ▲ is substituted for that of ; and dx therefore the different complete primitives of an equation of differences cannot be replaced by a single complete primitive. On the contrary, it may be shewn that out of the complete primitives corresponding to the component equations into which the given equation of differences is supposed to be resolvable, an infinite number of other complete primitives may be evolved corresponding, not to particular component equations, but to a system of such components succeeding each other according to a determinate law of alternation as the independent variable x passes through its successive values.

Ex. 1. Thus suppose the given equation to be

(Au)2 — (a +x) Au+ax= 0......

which is resolvable into the two equations

(4),

.......(5).

And suppose it required to obtain a complete primitive which shall satisfy the given equation (4) by satisfying the first of the component equations (5) when x is an even integer and the second when x is an odd integer.

The condition that Au shall be equal to a when x is even, and to x when x is odd, is satisfied if we assume

and it will be found that this value of u satisfies the given equation in the manner prescribed. Moreover, it is a complete primitive.

To extend this method of solution to any proposed equation and to any proposed case, it is only necessary to express Au as a linear function of the particular values which it is intended that it should receive, each such value being multiplied by a coefficient which has the property of becoming equal to unity for the values of x for which that term becomes the equivalent of Au, and to 0 for all other values. The forms of the coefficients may be determined by the following proposition.

PROP. If a, B, y... be the several nth roots of unity, then a* + B*+ y2... x being an integer, the function is equal to unity if x be equal to n or a multiple of n, and is equal to 0 if x be not a multiple of n.

n

+ √(− 1) sin the n roots will be

2π
n

=

n

Now if x be equal to n, or to a multiple of n, the above becomes a vanishing fraction whose value determined in the usual way is unity. If x be not a multiple of n, then since 1, the numerator vanishes while the denominator does not, and the fraction is therefore equal to 0.

Hence, if it be required to form such an expression for Au as shall assume the particular values P1, P2... P1 in succession for the values = 1, x = 2... x = n, again, for the values x = n + 1, x = n + 2 x = 2n, and so on, ad inf., it suffices to assume

where

...

▲u = Px-1P1+ Px-2P2 + Px-nPn.........
Au=

...

(7),

(8),

n

a, ß, y.

...

being as above the different nth roots of unity. The equation (7) must then be integrated.

It will be observed that the same values of Au may recur in any order. Further illustration than is afforded by Ex. 1, is not needed. Indeed, what is of chief importance to be noted is not the method of solution, which might be varied, but the nature of the connexion of the derived complete primitives with the complete primitives of the component equations into which the given equation of differences is resolvable. It is seen that any one of those derived primitives would. geometrically form a sort of connecting envelope of the loci of what may be termed its component primitives, i. e. the complete primitives of the component equations of the given equation of differences.

If x be the abscissa, u, the corresponding ordinate of a point on a plane referred to rectangular axes, then any particular primitive of an equation of differences represents a system of such points, and a complete primitive represents an infinite number of such systems. Now let two consecutive points in any system be said to constitute an element of that system, then it is seen that the successive elements of any one of these systems of points representing the locus of a derived primitive (according to the definitions implied above) will be taken in a determinate cyclical order from the elements of systems corresponding to what we have termed its component primitives.

2. It is possible also to deduce new complete primitives from a single complete primitive, provided that in the latter the expression for u be of a higher degree than the first with respect to the arbitrary constant. The method which consists in treating the constant as a variable parameter, and which leads to results of great interest from their connexion with the theory of differential equations, will be exemplified in the following section.

Solutions derived from the Variation of a Constant.

A given complete primitive of an equation of differences of the first order being expressed in the form