Sj/stems of Corpuscles describing Circular Orbits. 683 

 magnetic force was applied. Let ?/o -^o? 2/o> ^o ^e the values 

 of y, Zj -,j y J at this time, then when i = we have 



drj 



dK 



dt 



= ^o'-PI/o 



hence 



77 = ?o cos (ot-\- - (2/0 +p^o) sin cotj 



f = zq cos Qj^ + - (0(3 —pyo) sin o)^ ; 



and similarly -i 



.V = j7q cos (ot-\-— ^Q cos 0)^ ; 



the motion parallel to x is not affected by the magnetic 

 force. 



The values of y and z are given by 



y = 7j cos pt —^ sin j:)^, <? = 7/ sin j9i + ?'cos pt. 



The magnetic force a parallel to x calculated on the 

 assumption that the motion is steady is given by the equation 



__ f dz d 1 dy d 1 \ 

 ^^^{^tdy^r'^M d^' ? J 



dy 



where x\ y\ z' are the coordinates of the point P at which 

 the magnetic force is calculated. 

 Now, 



1 _ 1_/ _^ , d d\l ^ 



r'-'r Vdx'^ydi7'^^dJ']r^ 



dy' 



Substituting this value of 1/r' in the expression for a, and 

 expressing the coefficients of the differential coefficients of 

 1/r as periodic functions of the time, we can pass to the 

 solution when we take the acceleration of the particles into 

 account by the method explained at the beo-iunino- of this 

 paper. 



For the pur[)ose of discussing the magnetic properties of 

 the_ system under a steady magnetic field, the only terms 

 which are of importance are the non-periodic ones. Sub- 

 stituting the values of -r, ?/, 2-, -£, ~ in the expression for a, 



