Knowing u and v, we easily determine u' and v'. For, by equation (2),

2

= (α, − k1) e̟, cos(k ̧x + α ̧) + (α, − k ̧) c2 cos(k,x+ α.); similarly v' may be found.

In this case we have had an example of the integration of simultaneous differential equations, of an order higher than the first. We shall take another in

being two of the equations for determining the circumstances of the movement of a floating body in a position nearly of equilibrium, the other two being

we can eliminate z by multiplying the first by

the second by a, and adding, when we get

The value of z may be found from that of y; and if we know the initial circumstances, we can determine the arbitrary constants.

The same method may be extended to simultaneous partial differential equations, according to the principles developed in the Article 'On the Solution of Partial Differential Equations,' p. 62. Take, for instance, the equations

If we put 1-(cc') = m1, 1+√(cc') = m,, this may be resolved into factors,

d

d

d

2

{22 +m, ( a 2 + 8 ) } { // + m2 ( a 2 + 0) } = = 0;

dx

dy

the integral of which is

dx

dy

or

z = ɛ ̄m ̧1o þ(y — am ̧x) + € ̄12TM ↓ (y − am ̧x).

Mr. Airy, at p. 279, Note of the Undulatory Theory, gives as the equations for determining the small disturbances of an elastic medium in three dimensions

We might eliminate v and w, by cross multiplication, between these equations, and so obtain an equation in u which might be integrated, but it will be more convenient to proceed as follows. Let

We have now given a sufficient number of examples to enable the student to understand thoroughly the method, and we think that they show clearly the advantages of a process, which, to some persons, might appear to carry out to a startling extent the principles on which it is founded.

GEOMETRICAL THEOREM.*

3

LET  ̧ ̧....... be a polygon of n sides inscribed in a circle, a1, α„, α„, &c. the angles which the sides 4‚Ä  &c. subtend at the centre.

3)

Then

27

37

If n be even, adding all these together, we get

or the sum of the alternate angles is equal to n−2 right angles, a curious extension of Euclid, 111. 22.

* Cambridge Mathematical Journal, Vol. I., p. 192,