and the Dynamical Theory of Diffraction. 419 



where 



ax ay dz p dp 1 r 



F =-c 2 F 1? &c. 



The remaining derived vectors are simply repetitions of 

 these with only change in the constants. 



These derived vectors can represent the vector potential, 

 the electric displacement, the electric current, the magnetic 

 displacement, or the magnetic force in the electro-mngnetic 

 theory, and the displacement, the velocity of the particles, or 

 the rotational displacement or velocity of rotation, on the 

 elastic-solid theory. 



The most interesting case is that of symmetry round an 

 axis, say X. In this case 



d n 1_ 1 (-1)" 

 dx n p ~~ p n 1 . 2 . 3 . . . n 



where Q n is a zonal surface harmonic. 



If a is the angle measured around the axis of x from y, and 

 that made with the same axis, and we put 



p sin = r, 



we can then write, when m is even, 



V -— f-± \-i±i Q , 



J n.2 , TJ2 



m — ^ ^* mi ■ L - L mj 



N w+ i— V G 2 m +l + H m _j_i, 



M m+ i = N m+ ir, 



1 dM m ^ 

 r dr 



F m =-A 2 F m _ 2 =+~ — — , 



-p A2T) I^m-2 1 dM m _i 



K m _-A±( m _ 2 __--_ g _, 



F w+ i=0, 



r 1 6/^ 2 cZr 2 r* dr ) 



The equation of continuity becomes 



d(F m r) dCR m r) =Q 

 dx dr 



These can be expressed in terms of p and 0, as follows : 



