272 Prof. Townsend and Mr. Morrell on Electric 



and he concluded that, when one end of the solenoid was con- 

 nected to earth and the other end insulated, the frequencies of 

 the free oscillations were proportional to the numbers 1, 3 ? 

 5, 7, etc. More recently the wave-lengths \ x and X 2 of the 

 first and second normal modes of oscillation of an insulated 

 solenoid have been investigated bv Lenz *, who found that 

 the ratio 2\ 2 /\x depends on the ratio of the length I to the 

 radius r of the solenoid. For small values of l/r, 2\ 2 /~^i is 

 less than unity ; as l\r increases, 2X 2 Ai also increases and 

 attains a maximum value 1*21 when Z/r=4. For larger 

 values of Ijr, 2\ 2 /\ 1 approaches unity. Leide f has investi- 

 gated the frequencies of the normal oscillations of solenoids, 

 and has compared his determinations with a formula given 

 by Siegbahn J, in which the capacity to earth is taken into 

 consideration. The experimental determinations do not 

 appear to be in general agreement with the formula. 



8. In the following calculations approximate methods are 

 used to determine the frequencies of the normal modes of 

 oscillation of a solenoid, the length of the solenoid being 

 much greater than the diameter, and .the distance between 

 the nodes measured along the axis also large compared with 

 the diameter. The solenoid is supposed to be remote from 

 objects which would affect its capacity. The wave-length is 

 taken as the wave-length in air, as measured in the ordinary 

 way by a wave-meter. 



Let i be the current, v the potential, and e the charge per 

 unit length at any section of the solenoid at a distance x 

 measured along the axis from one end, S the self-induction, 

 R the resistance, and C the capacity per unit length. 



In a free oscillation the variables i, v, and e are connected 

 by the relations 



s S^=-£ a) 



and 



dv _de _ di 



Kj Tt-dt~~T^ (2) 



w T hen the rate of propagation along the axis of the solenoid 

 is small compared with the velocity of light in air. 

 If R is neglected, the equation for i becomes 



* W. Lenz, Ami. d. Phys. iv. 43, p. 749 (1914). 



f Leide, Arkivfur Mat. Ast. Phys. xii. no. 24 (1917). 



% M. Siegbahn, Arch.f. Electrotechnik, iv. p. 305 (1916). 



