Now the fact of considering that the force in any direction is the resolved part in that direction of the whole force put in play, requires that the forces be all of the same nature: how does it happen then that a part of them may be omitted altogether? Should it be urged in reply, that the motion of a particle in a given direction is not affected by a force which acts always at right angles to that direction; I answer that this is not the solution of the real difficulty, though most persons appear perfectly satisfied with it. That the absolute motion of the particle will be such as continually to change the plane in which it moves is quite obvious. If then, as M. Fresnel supposes, the velocity depends on the position of this plane, the velocity itself must be continually varying for the same ray.

Nor has the plane in which the particle moves a reciprocating motion. The construction consisting of an ellipsoid cut by the plane of vibration through its centre sufficiently proves this; for it is found that the whole force due to a displacement in one of the axes of the elliptic section acts in the direction of a normal to the ellipsoid at the extremity of that axis. Suppose then the particle to be at its greatest distance from its position of rest; the action of the normal force causes it to return in a direction above the plane of its disturbance, (suppose). When it has reached the other extremity of its oscillation, the force tends to pull it below the line of its return : by each action, therefore, the change from its original line of motion is in the same direction, and this will take place continually, so that the plane of motion will continually vary, and the velocity of transmission constantly increase or constantly diminish.

These points appear to me weak points in the theory: the former is indeed of such magnitude, that were there nothing to limit its effects, the results would be very far from the truth. The error, however, which is committed by this step is exactly righted by the second, and thus two hypotheses which individually are erroneous, do, when combined, lead to correct results. Indeed it is manifest that whereas the former error arises from not giving to the front of the wave its due effect, the latter arises from giving it an effect which

it could not produce: the former requires that the force should act out of the plane of the wave, the latter rejects the part which does; and these will right each other if we can shew (as I trust I have done in the sequel) that the actual vibratory force is in the front of the wave. I could have desired that my investigations should have assumed a more inviting form, but I have not the means at present of throwing them into a shape other than that under which they

appear.

The first step I have taken is to prove the transversality of the vibration, and thus having established a direction in which vibrations do take place, I suppose that the forces put in play by a displacement may be determined (as far as their action in the direction of that displacement alone is concerned) in the same manner as Fresnel does. The modification then which I propose, consists in restricting the theorem of Fresnel, and reducing it to the following:

"That the whole vibratory force put in play by a displacement in a direction which admits of a vibration, is the sum of the resolved parts along that direction of the vibratory forces due to the resolved parts of the displacement along the axes of elasticity." elasticity." With a direction normal to those of vibration I have nothing to do, except to prove that the force in that direction is not part of the vibratory force.

SECTION II.

Investigation of the Motion of a System of Particles within a Crystal.

1. WE shall assume that the arrangement is an arrangement of symmetry with respect to three planes mutually at right angles to each other.

Let their lines of intersection be taken as the axes of co-ordinates.

x, y, the co-ordinates of the particle under consideration in its position of rest.

x + a, y + ß, ≈y its co-ordinates after the time t

x+dx, y+dy, +8% the co-ordinates of another particle whose distance from the former is r.

Then, by pursuing a process precisely analogous to that which applies to non-crystallized media, (Trans. Camb. Phil. Soc. Vol. VI. Part I. page 162.) supposing the particle in vibration, we have the following equations of motion:

διαδη
sin? kop

d'a

dt2

2

2

multiplying the equations by P, Q and R respectively, and adding them, we may put the result under the form

(a — A) (b − A) (c- A) – X2 (a− A) – Y2 (b− A) – Z (c-A)-2XYZ=0,

-

or (A − a) (A −b) (4 −c) – X2 (A − a) — Y2 (A − b ) − Zo (A – c)

+2XYZ=0, (2);

an equation which it may be readily shewn gives three possible values

of A.

2. These values are either two positive and one negative, or two negative and one positive, for if we write the equation under the form

A3 − (a + b + c) A2 + {ab + ac + bc − (X2 + Y2 + Z3)} A

- abc + a X + b Y2 + cZ2 + 2XYZ=0,

it will be evident that the coefficient of A is equal zero (1).

The roots then must assume the form 4,4 − (A, + A2), in which AA, may be either both positive or both negative: suppose the former.

3. Now, corresponding to any value of A, a value of P, Q, R, respectively can be determined; but A is the velocity of transmission of a vibration whose direction makes with the co-ordinate axes the angles cos1 P, cos1 Q, cos' R respectively, and which is transmitted in a direction making with the same axes other angles 0,

and y.

We conclude then, that there are in general two directions and no more, in which a vibration taking place, the transmission along a given line is possible. A disturbance in a given direction being resolved into these two, will give rise to two different rays, transmitted with different velocities.

4. The third value of 4 which is negative, will not correspond to a vibration; the manner in which it may affect the motion, and the probable results to which it gives rise, I have fully discussed in a paper read before this Society a short time since, and shall leave it untouched in the present Memoir.

5.

Discussion of the Equation for A.

As a preliminary step towards a complete discussion of this equation, we will first consider the medium perfectly symmetrical.

Transform the co-ordinates in such a manner that the axis of x' shall coincide with the direction of transmission, and that of y' lie in the plane of xy.

Denote the angle between the axes of x and a' by the symbol (xx), and so on for the others: