14 The Hon. J. W. Strutt on some Electromagnetic Phenomena. 



Again, supposing x x x 2 both given, we must have, when E is 

 a minimum, 



d[E dE 



dx 3 dx 4 '" } 



and thus 



2E (min.) =^[(1 l)a? x + (1 2)*J 



+^[(12)*, + (2 2)*»+(2 3)*J 



= [1 1 )aj + 2(1 2)*, x 2 + (2 2)a* + (2 Z)x 2 x 3 . 

 As before, 2E might have been 



(11)^ + 2(12)^ 2 +(2 2)^ ; 



and therefore the minimum value is necessarily less than this, 

 and accordingly the signs of x 2 and x 3 are opposite. This process 

 may be continued, and shows that, however long the series, the 

 initial induced currents are alternately opposite in sign. In any 

 definite example, the actual values of the initial currents are to 

 be found from the solution of the linear equations 



but the sign of the result does not appear at once from the form 

 of the expression so obtained. In order to exhibit it, it is ne- 

 cessary to introduce a number of relations which exist between 

 the induction-coefficients, and which are the analytical expres- 

 sion of the fact that the energy is always positive, whatever may 

 be the values of a? 2 , x 3 , . . . 



It has been assumed throughout that the time of rise or fall 

 of the current in the primary wire is very small as compared with 

 the time-constants of the other circuits. In the case of coils, 

 such as are generally used in induction-experiments, and which 

 are not clogged by great external resistances, this condition is 

 abundantly fulfilled at the break of the voltaic current*. The time 

 of rise depends more on the nature of the circuit, but may be 

 made as small as we please by sufficiently increasing the resist- 

 ance in proportion to the self-induction ; of course, in order to 

 get an equally strong current, a higher electromotive force must 

 be employed. In this way the rise may be made sufficiently 

 sudden to fulfil the condition. Indeed, with a battery intense 

 enough the rise of the current at make may become more sudden 



* A rough measurement by Maxwell's method (Phil. Trans. 1865) gave 

 for the time-constant of the circuit composed of the two wires of coil A 

 *0023". The time-constant is the same whether the wires are collateral or 

 consecutive, the greater self-induction of the latter arrangement being 

 balanced by its greater resistance. For one wire only, the time-constant 

 would be half the above. 



