Mr. G. G. Stokes on the Aberration of Light. 11 



which corresponds to the point (j!',y, «) in its front at the time 

 t, we have 



.^' = 5; + (to + V)dt\ 



and eliminating .r, y and z from these equations and (1.), and 

 denoting t, hyj'{x,i/, i), we have for the equation to the wave's 

 front at the time t + dtj 



z' -{w + N)dt=C + Yt 



+/{^ - (« - v^^) dt, y - (. - vg) dt, t}, 



or, expanding, neglecting dt'^ and the square of the aberra- 

 tion, and suppressing the accents of sc, y and a, 



s = C + Yt + \;, + {w + Y)dt. . . . (3.) 



But from the definition of ^ it follows that the equation to the 

 wave's front at the time t -\- dt will be got from (1.) by put- 

 ting i + d t for tf and we have therefore for this equation^ 



z=C + Yt + i;+[Y + ^Jdt. . . . (4.) 



Comparing the identical equations (3.) and (4.), we have 



7^ = TO. 

 d t 



This equation gives ^= / tadt: but in the small term ^ 



we may replace / wdt ^y =rr / lodz: this comes to taking 



the approximate value of z given by the equation z = C + Ytf 

 instead of /, for the parameter of the system of surfaces formed 

 by the wave's front in its successive positions. Hence equa- 

 tion (1.) becomes 



z=C + Yt + y/*'^^^' 



Combining the value of ^ just found with equations (2.), we 

 get, to a first approximation, 



TT 1 /*dw , - TT 1 PdW , ,^ . 



2 YJ dx ' ^ 2 VJ dy ' ^ ^ 



equations which might very easily be proved directly in a more 

 geometrical manner. 



If random values are assigned to u, v and to, the law of aber- 

 ration resulting from these equations will be a complicated 



