1885.] equations of vibrations of ether in a light wave. 283 



The first terms will evidently be obtained by differentiating 

 only the trigonometrical parts. Thus 



c i = - f (% - K z )> d i = f ( a sV ~ a A 



and similarly we may deduce the other terms. 

 For example, 



where a, b', c', a" stand for the second terms in A, B, C, D. 



If we proceed to operate upon f, rj, £ we may obtain a second 

 derived vector | 2 , y. 2 , £ 2 which will have the form 



tj^-ttp^U^-x -J p 3 -jsmp{r-at) 



(.. b.x+bjy + bjs\ , ,J 



+ f & x — a? — * f ^- J cos p (r — at)} , 



where 22 relates to the summation for different values of p and 

 also for the completion of the coefficients of the several terms. 



This will become ff 2 = — 2p 2 |f , provided 

 a x x + a 2 y + a z z = 0) 

 &^ + 6 2 y + ^ = 0} ^ ; ' 



7 5- 7 ^-7 V 



which form the conditions that -P- + -r-° + -r 5 = 0. Operate upon 



efe ay dz 



the first of these with y 2 ; write a' and b' for the second terms in A 



and 5, and observe the equations (3) by which these are to be 



found. 



2 , 2 (da. , c/\ , fZa \ n 



1 ^ 2 3 p ^ dx dy dz) ' 



Now this is of the order \/r compared with a and b, and there- 

 fore to this degree of approximation we may say that 



#£ + yv + z% = o, 



or that the displacement lies in the wave front (Stokes' papers, 

 Vol. II., On the Dynamical Theory of Diffraction, Art. 27, p. 275). 



A proof that x% + yv + z£ is approximately zero, on the con- 

 dition that -j- + -j- + -^ = may be given in this way, 



v 2 (#£ + yv + %K) = «v 2 £ + yv 2 v + *v% 



