Prof. Challis on the Dispersion of Light, 461 



is wholly rectilinear, the number of the rectilinear axes must be 

 conceived to be so great that the transverse motions are mutually 

 destructive. We may, if we please, suppose the evanescence of 

 the transverse motions to result from the motion being com- 

 posed of an unlimited number of such motions in cylindrical 

 threads as those which are shown above to be physically possible* 

 the axes of the cylinders all passing through the centre. This 

 being understood, let us consider by itself the motion within a 

 pyramidal space having its vertex at the centre, and its vertical 

 angle indefinitely small. If V be the velocity and a the conden- 

 sation at the distance r from the centre, we shall have, as in the 

 case of plane waves, to the first order of small quantities, 



da dN 

 Ka 'dr + dt U * 



Also if the lines of motion, that is, the lines drawn always in the 

 direction of the motion of the particles through which they pass, 

 all converge to a moving centre, the same equation still applies 

 to the motion within each small pyramidal space, although the 

 total motion in this case may not be alike in all directions from 

 the centre. And generally, supposing the lines of motion and 

 surfaces of displacement to be of any form whatever, and either 

 to be fixed in space, or to vary in position with the time, pro- 

 vided there is no abrupt change of direction in passing at a 

 given instant from a point on one line to an adjacent point of a 

 contiguous line, or from point to point of a given line, and no 

 extraneous force acts, we have 



da dV 

 Ka 'ds + dt ~ U ' 



the line s being measured along a line of motion from a given 

 origin on the same, and V being the total velocity. These 

 theorems are stated here because they will be employed subse- 

 quently ; but for their demonstration I must refer to my pre- 

 vious hydrodynamical researches, especially those contained in the 

 Numbers of the Philosophical Magazine for February and Novem- 

 ber 1853. 



Returning now to the case in which the lines of motion are 

 rectilinear and pass through either a fixed or a moving centre, 

 and joining with the foregoing equation that of constancy of 

 mass to the same approximation, namely, 



da dY 2V n 



di + -¥ + T=°> 



