160 Mr KELLAND, ON THE DISPERSION OF LIGHT, 



0, <p and ^// being the angles which a straight line makes with the 

 co-ordinate axes of .r, y, k. For simplicity put 



X cos 6 + 1/ cos <p + % cos ^ = p, 

 p being the projection on the line making angles 0, (p, ^ with the 

 axes, of the distance OP of the point P from the origin: the above 

 equations then become 



a = a eos/c {nt — p), 



fi = b cosJe {nt - p), 



7 = c cos A {nt — p). 



It is true a more general solution of these equations would have been 

 obtained by assuming the values of k, 6, (p, -^ different for each of 

 the three equations, but as the complete solution will consist of a series 

 of terms similar in form to the above, it is sufficient for our purpose 

 to exhibit that term above which has the same period for each of the 

 three directions, and which consequently corresponds to one and the 

 same undulation. 



From these equations it is manifest that the same state of motion 



2 TT 



recurs when hp is increased by Swtt, and consequently -y- is the length of 



a wave; also the motion at the end of /+ AMs the same at the point 

 p + Ap as at the time t at p, when nAt = Ap, whence the velocity of 



transmission parallel to jo = -^ = re. 



It may be worth while to notice here that the proposition which 

 we have considered assumes the velocity of transmission to be the same 

 in all directions : in general, however, this will not be the case, the 

 direction of transmission being defined by the simultaneous transmission 

 of a system of waves, and the velocity will have reference to that 



direction only; but as -f- is independent of any particular direction, and 



depends only on the nature of the substance, it must be either the 

 velocity of transmission itself, or the velocity in a direction making 

 a constant angle with that of transmission, and consequently varies as 



