588 Professor CHALLIS, ON A MATHEMATICAL THEORY 



>|/ = 2 \fin cos — (z - at y/ 1 + — j- + c)} 



-J^ = M^ = 2 { /m sm — - (^ - a< V 1 + — r + c) } 



_ _.^ =os = 2{/m, y/ 1 + sin — (xf-a^ V 1 +-T + 0}- 



a at TT A T 



It follows, since in each of the terms under the sign 2 the quantities which are independent 

 of g and f are at our disposal, that we may satisfy by this integral any state of the fluid in 

 the direction of z, subject to the limitation that the condensation and velocity are at all times 

 small. The course of the reasoning shews that the particular form of the function G which 

 has conducted to the above results, has not been adopted as an analytical artifice, but is really 

 the only form which determines the velocity of propagation, and gives a definite solution of 

 the Problem. The particular kind of motion it represents, and the component character of the 

 whole motion as consisting of an indefinite number of such motions, are accordingly to be regarded 

 as physically true. These results explain the fact of the composition of light. 



6. Before proceeding farther, it will be worth while to compare the foregoing investigation 

 with that which I have given in my Paper on Luminous Rays. (Cambridge Philosophical Trans- 

 actions, Vol. VIII. Part III. p. 363). It may be remarked, that the two investigations agree in 

 their results, but differ in the course of the reasoning. In the Paper referred to, the velocity of 

 propagation is assumed to be uniform (p. 365), and the form of the function expressing the nature 

 of the vibrations is deduced from this assumption (p. 368). In the present communication the 

 form of that function is first obtained by o priori reasoning from the Hydrodynamical Equations, 

 and the uniformity of the rate of propagation is then strictly deduced. The inferences in the 

 former Paper (p. 365), drawn from the supposition that the velocity of propagation is uniform 

 when the motion is not small, still hold good. It may also here be remarked, that the consider- 

 ations in p. 366 on which the arbitrary quantities c, c', c" were made to vanish, are superseded 

 by the more general reasoning in Art. 1. of this Paper. 



7. I proceed now to the consideration of equation (12), viz. 



-/r,+-ri +4e/=0. 

 dx- ay 



As this equation does not contain t, there is no propagation of motion in any direction 

 parallel to the plane of xy ; or, the propagation in the direction of z takes place without 

 lateral spreading. A value of / expressed in finite terms is not therefore required, as in the 

 case of the integration of equation (11), for deducing velocity of propagation. It may however 

 be argued, that as a particular form of (p was found, by which the vibrations in the direction 

 of z were defined, prior to any consideration of the manner in which the fluid was put in motion, 

 so a particular form of / exists by which the condensation and velocity in directions transverse to 

 the axis of z are defined, and which is equally independent of the arbitrary disturbance. As 

 this form may, or may not, be capable of expression in exact terms, I shall first apply to 

 equation (12) the process already applied to equation (11), for the purpose of ascertaining 

 whether any exact value of the integral satisfies the conditions of the Problem. 



8. The equation (11) coincides in form with (12) by putting - 1 for a', and 4e for h'. 



That is, since e = — , we shall have - e in the place of e in the integral of (I2). Hence 



