582 Lord Rayleigh on the Induction- Coil. 



in the manner of break, which usually occupies too long a 

 time, or at least departs too much from the ideal of an 

 instantaneous abolition of the primary current. A third 

 complication arises from the capacity of the secondary coil, 

 in virtue of which the currents need not be equal at all parts 

 of the length, even at the same moment of time. If we 

 ignore these complications, treating the break as instan- 

 taneous, the iron as ideal, and the secondary as closed and 

 without capacity, the theory, as formulated by Maxwell *, 

 is very simple. In his notation, if x, y denote the primary 

 and secondary currents, L, M, N the coefficients of self and. 

 mutual induction, the energy of the field is 



iLtf 2 + M^+|% 2 (l) 



If c be the primary current before the break, the secondary 

 current at time t after the break has the expression 



M 



N 



y = c e-**-', (2). 



S being the resistance of the secondary circuit. The 

 current begins with a value c . M/N, and gradually disappears. 

 The formation of the above initial current is best under- 

 stood in the light of Kelvin's theorem, as explained by me 

 in an early paper f. For this purpose it is more convenient 

 to consider the reversed phenomenon, viz., the instantaneous 

 establishment of a primary current c. The theorem teaches 

 that subject to the condition x = c the kinetic energy (1) is to 

 be made a minimum ; so that 



Mc + %=0 



gives the initial secondary current. In the case of the break 

 we have merely to reverse the sign of y. 



Immediately after the break, when # = and y has the 

 above value, the kinetic energy is 



i^tf, or £— • 



Immediately before the break the kinetic energy is ^Lc' 2 , so 

 that the loss of energy at break — the energy of the primary 

 spark — is 



LN— M 2 



* "Electromagnetic Field," Phil. Trans. 1864; Maxwell's Scientific 

 Papers, i. p. 546. 



t " On some Electromagnetic Phenomena considered in connexion with 

 the Dynamical Theory," Phil. Mag. xxxviii, p. 1 (1869) ; Scientific 

 Papers, i. p. 6. 



