Oscillations in Straight Wires and Solenoids. 273 



which is satisfied by values of i of the form 



i — A sin pt sin qx t ■ . (4) 



when p and q are connected by the relation 



S(> 2 = q* (5) 



With an insulated solenoid the current is zero at the ends, 

 x = and x = l, so that the values of q must satisfy the con- 

 dition sin ql = 0. Thus in the normal modes of oscillation 

 represented by equation (4) the values of q are 



9i = 



IT 



9* 



T' 



73 



3tt 



T' 



etc.. 



and the distances between the nodes measured along the axis 



are 



I, 1/2, 1/3, etc. 



The corresponding values of p which are proportional 

 to the frequencies are obtained from equation (5). When 

 the distance between the nodes is long compared with the 

 diameter of the solenoid, the quantity S is nearly the same 

 as the self-induction per unit length obtained with a uniform 

 current ; so that S is approximately equal to 47r 2 N 2 a 2 , N being 

 the number of turns per unit length and a the radius of the 

 solenoid. 



The capacity C depends on the distance between the nodes 

 as measured along the axis, and it is necessary to find the 

 value of C corresponding to each value of q. 



Let the distribution of the charge e along the solenoid be 

 represented by the curve AB, fig. 2. If x be measured from 



Fisr. 2. 



Ky 



the section at P where the charge has a maximum value e > 

 the charge at the same instant at a distance x from P is 



2ttx 

 <? = ^ cos-^-, h being twice the distance between the nodes. 



