XLI. A Mathematical Theory of Luminous Vibrations. By the Rev. J. Challis, 

 M.A., F.R.AS., Plumian Professor of Astro7iomy and Experimental Philo- 

 sophy in the University of Cambridge. 



[Read March 6, 1848.] 



In three preceding communications to this Society I endeavoured to explain some of 

 the principal phaenomena of Light on the Hypothesis of Undulations, regarding the aether as 

 a continuous and elastic fluid, and applying to it the usual Hydrodynamical Equations. I 

 propose now, on the same principles, to investigate the particular nature of the aetherial vibrations 

 which produce light, and the laws of their propagation under given circumstances. As this com- 

 munication is intended to be supplementary to the three former, I shall take occasion to advert 

 to any reasoning they contain, to which I may be able to add elucidation or confirmation. 



1. Let a' {\ + s) be the pressure at any point xyz of the aether at any time t, s being 



a small numerical quantity, the powers of which above the first are neglected ; and let u, v, w, 



be the resolved parts of the velocity at the same point and at the same time, in the directions 



of the axes of co-ordinates. Then, retaining only the first powers of ti, v, w, we have, as is known, 



., ds dii , „ ds dv , ^ „ ds dw 



"'■T.^dt^'^^'^ "'-dy^-dt-'^^'^ ''■r.^di = ''^'^ 



, ds du dv div , , 



^"•^ :77 + l~ + J~+l~ =" W- 



dt dx ay dz 

 The last of these equations gives by means of the other three, 



d's „ fd's d's d's\ , ^ 



d?-''-fc+rfy^^d?)=° ('^- 



Suppose, for the moment, that s has been obtained from this equation by integration. Then 

 for the velocities we have, 



o r^* J, 2 d. fsdt 



u = c - a- / dt = c - a"^ . — ^ (o), 



Jdx dec 



" rds , , „ dfsdt , ^ 



v = c -a' —~ dt =c - a- . — - — (7), 



J dy dy 



w = c -a- -—dt=c - o . — -; — (8;, 



J dz dz 



wnere v, c, and c" are functions of co-ordinates only. It is to be observed that these values 

 of u, V, w are perfectly general, being obtained prior to any consideration of the way in which 

 the fluid was put in motion, and consequently apply to all points of the fluid in every instance of 

 motion in which powers of the velocity and condensation above the first may be neglected. Now 

 the motions of the aetherial medium are vibratory, or, at least, not permanent. There is no known 

 cause to produce motions in the aether, which either wholly or in part remain permanently the 

 same at the same points of space for any length of time. And even if, from causes with which 



