receives the same motion from the change of tension, and therefore

2

and therefore, putting n2 for, equation (12) becomes

This equation being no longer linear, cannot be integrated as the preceding equations were, and we must therefore have recourse to an approximate solution. If we suppose c to be small, we may substitute in that term the value of x derived from the supposition of c=0. That gives us

and therefore

x= A cos (nt + B),

[cos{(n+at+B+ ß} + cos {(n − a) t + B−ß}].

The most important conclusion from this result is, that if A=0, or the ball be originally at rest, it will have no motion communicated to it by the vertical motion of the point of suspension: and that if it has an oscillatory motion in any direction, this will not be permanently altered unless a=2n, or the period of oscillation of the pendulum be double of that of the point of suspension. In this case the integral becomes infinite; and if, as before, we put it into another shape, we find that x increases continually with the time. This accords with experiment, which shows that the arc of vibration of a pendulum may be increased indefinitely by giving the point of suspension a vertical motion of oscillation, the period of which is half of that of the pendulum.

ON THE EVALUATION OF A DEFINITE

MULTIPLE INTEGRAL.*

IN a memoir read before the Academy of Sciences of Paris, and inserted in the Comptes Rendus, Vol. VIII., p. 156, M. Lejeune Dirichlet called the attention of mathematicians to the remarkable multiple integral

which is to be taken between the positive limits of the variables determined by the inequality

the number of variables, and therefore of integrals, being any whatever. The result at which M. Dirichlet arrives

I being the second Eulerian Integral.

The actual calculation is not given in the paper referred to, though the process is indicated; but M. Liouville has investigated the value of the integral by a method different from that employed by M. Dirichlet, and his memoir (Journal

* Cambridge Mathematical Journal, Vol. 11., p. 215.

de Mathématiques, Vol. IV., p. 225) is a very elegant specimen of analysis. The integral itself deserves attention, not only as being a remarkable analytical extension of that property of the first Eulerian Integral by which it is connected with the second, but also because it frequently occurs in the investigation of areas of curves, contents of solids, centres of gravity, and other physical and geometrical problems of a similar kind. From its extensive application to cases which are of such frequent occurrence, this multiple integral ought to receive a prominent place in elementary works on the Integral Calculus, and on that account I here bring it before the English reader. The method by which I propose to evaluate this integral is, I believe, new; and I am anxious to show its application in this case, not only because it exhibits very distinctly the nature of the connexion between this integral and the function F, but because it can also be applied with great advantage to the calculation of a number of other definite integrals. In the present paper, however, I shall confine myself to the integral of M. Dirichlet, and a more general one of the same kind which is given by M. Liouville.

In the first place, following the method of M. Liouville, we shall transform the integral so that the limits shall be of the first degree only. This is easily done by assuming

(-)'=x', ()'-', (-)"'-', &c.,

a

from which

α

dx

p

تھا

=y',

dz Y dz',

da== '"da", dy=y\"dy, da=7a-da, &c.

On substituting these values for the variables and their differentials, the integral becomes

the limits of the variables being given by the inequality

and, dropping the accents which are no longer necessary discrimination, we have to calculate the integral

-1 m-1_n-1

= Sdx Sdy [dz... - - ... ...........(6):

the limits being given by the inequality

x + y + z... 1.

for

If the variables be only two in number, x and y, the integral is reduced to

since the limits of y are 0 and 1 − x.

The evaluation of this integral, by a method due to Professor Jacobi, may be found in the Cambridge Mathematical Journal, Vol. I., p. 94, and it is by an extension of that method that M. Liouville. has calculated the general integral under consideration. Instead of employing it we shall proceed in the following manner:

Then, as a varies when y, z... are constant, dx=dv, and (10) becomes

U=

U = |dv

m-1.

(v — y − z...) 11........(7).

We might now integrate with respect to v, but, for the convenience of our future operations, we shall only indicate the operations. The extreme limits of v are 0 and 1, and we may therefore write

Now by the symbolical form of Taylor's Theorem we have

n-1 2

d

dv

m

d

2

1-1

U = ['* dv [ de " ... fdy y "; " (v — ——...) ̃........(10),

the limits of y being 0 and v

-2

To find the limits of t we have recourse to the following considerations. Supposing, for simplicity, that there were only two variables, x and y, we have

Now, when y = 0 the first side becomes v1; and in order that the second side should be reduced to that form, we must have t=0. Again, when y=v the first side becomes zero, 7 being positive; and in order that the second side may also become zero, we must have t∞.