Dynamical Theory of Di fraction. 231 



above result will be correct, only on the supposition that the 

 direction of vibration is perpendicular to both r and r x . 



If this is not the case, we must resolve the displacement 

 (which is necessarily perpendicular to r^) in the direction 

 perpendicular to r (in the plane of r and the vibration). In 

 doing so, we observe that if u', t/, w' be the components of 

 the velocity perpendicular to r, and u, v, w those perpen- 

 dicular to r 1} then 



v! = u — ql, &c, 



where q is the component of u, v, w along r. 



In the equation (4), then, u must be changed into u Q —q l, 

 a corresponding change being effected in the second term. 



The modified equation will then be 



i „ s= ^ = qi dS + (^o_ l ^o\dS _ _ ( 



r \or or J r K J 



where 



and I, m, n the direction cosines of r. 



Taking now the particular case of a displacement along 



the axis of z, viz. f \t — -) (the direction of propagation 



along the axis of x), we have, neglecting terms depending 

 on r~ 2 , and remembering that the resolved part of the 

 displacement perpendicular to r is f\t') sin <£, where (j> is the 

 angle made by r with the axis of z, from (14) and (15) 



if 9 is the angle between r and the direction of pro- 

 pagation, both measured in the same direction. 

 In particular, if 



,/ a?\ . . 2irc / a?\ 

 „,. 1 A 2tto 1 2we(. r\ 



which is Stokes's result. 



