Oscillations of various Periods. 375 



assume the value (^M') given by ordinary hydrodynamics, 

 in which viscosity is not regarded. 



The use of the principal coordinates would not often be 

 advantageous when the object is a special calculation of 1/ 

 and E/, rather than the establishment of a general theorem. 

 In one very important case — that of two degrees of freedom 

 only — the question does not arise, since but one other coordi- 

 nate ^r 2 enters in addition to ^rj. Under this head we may 

 take the problem of the reaction upon the primary circuit of 

 the electric currents induced in a neighbouring secondary 

 circuit. In this case the coordinates (or rather their rates 

 of increase) are naturally taken to be the currents themselves, 

 so that ^ is the primary, and yjr 2 the secondary current. 



In usual electrical notation we represent the coefficients of 

 self and mutual induction by L, N, M, so that 



and the resistances by E and S. Thus 



a n = L, a 12 = M, a 22 — N" ; 



b n = R, b 12 = 0, b 22 = S ; 

 and (6) and (10) become at once 



» 2 M 2 S 



E '= E+ #T?F> < u > 



u — L 8 2 +yW • • • • • W) 



These formulae were given long ago by Maxwell*, who remarks 

 that the reaction of the currents in the secondary has the effect 

 of increasing the effective resistance and diminishing the 

 effective self-induction of the primary circuit. 



If the rate of alternation be very slow, the secondary 

 circuit is without influence. If, on the other hand, the rate 

 be very rapid, 



M 2 S LN-M 2 



The formulae (11) and (12) may be applied to deal with a 

 more general problem of considerable interest, which arises 

 when the secondary circuit acts upon a third, this upon a 

 fourth, and so on, the only condition being that there must be 

 no mutual induction except between immediate neighbours in 



* Phil. Trans. 1865 j M is misprinted for M 2 . 



