Quantitative Study of the High- Frequency Induction- Coil. 513 



to the phenomena of the varying electrostatic field ; while 

 some have attempted to follow the behaviour of the current in 

 one of the circuits. Colley * has accomplished this by making 

 the period very slow and using an " oscillometer " — either a 

 fluctuating flame or the mirror of an exceedingly sensitive 

 galvanometer. 



Theoretical. 



The mathematical theory of the so-called Tesla coil has 

 been discussed by Oberbeck and others f. The discussion 

 which follows is a modification and extension of Oberbeck's. 

 It is essentially a discussion of a system with two degrees of 

 freedom. 



Let the suffixes 1 and 2 refer to the primary and secondary 

 circuits respectively. Let V represent the difference of 

 potential at the terminals of the condenser, I the current, 

 Q the charge on the condenser. L self-induction, M mutual- 

 induction, K capacity, R resistance. We have then to 

 distinguish two cases : — (1) The secondary circuit is open ; 

 that is, is closed by a capacity. (2) The secondary circuit is 

 closed by a resistance, or short-circuited. 



(1) The differential equations of the system are : — 



V 1 +L 1 ^+M^ 2 + R 1 I 1 = ; 



dt dt 



dh+M— 1 



dt ^ dt 

 Making use of the equations of continuity, 



V 2 + L 2 ~? + M ~! + R 2 I 2 = 0. 



d.Q 



1 ~ dt 9 

 and of charge, 



v - K , 



these become 



_ + L 2 ^- + M^-+R 2 ^_0. 



* Wied. Ann. xxvi. p. 432 (1885), xxviii. p. 1 (1886), xliv. p. 102 

 (1891). See also Hotchkiss and Millis, Phys. Rev. iii. p. 49 (1895). 



t Oberbeck, Wied. Ann. lv. p. 623 (1825). .Domalip and Kolacek, 

 Wied. Ann. lvii. p. 731 (1896). Bliimcke, Wied. Ann. lviii. p. 405 

 (1896). Wien, Wied. Ann. lxi. p. 151 (1897), gives the mathematical 

 theory for two circuits in the general case, where they have not only 

 mutual induction, but also mutual capacity and " mutual resistance " — 

 that is, affect the distribution of the current in the bodies of the 

 conductors. 



