Self-induction of Wires, 67 



conditions ((yd), saving a great deal of preliminary work. But, 

 on coming to practical cases, in which the inductances do 

 admit of being definitely localized in and between the six 

 branches of the Christie, we have to expand the ins properly, 

 using (5d) or as much of them as may be wanted, and so 

 obtain the various results in Part VI. Therefore equations 

 (tid) are only useful as a short registration of results, subject 

 to (5c/), and in the remarkably short way in which they may 

 be got ; a method which is, of course, applicable to any net- 

 work, which can only have as many independent inductances 

 as there are independent circuits, plus the number of pairs of 

 the same. 



(4) Reduction of Coils in Parallel to a Single Coil. — On 

 page 186, Part VI., in speaking of the inductometer, I referred 

 to the most useful property that a pair of equal coils in parallel 

 behave as one coil to external impressed force, whatever be 

 the amount of mutual induction between them ; a property 

 which, excepting in the mention of mutual induction, I had 

 pointed out in 1878*. But, although there appears to be no 

 other case in which this property is true for any value of the 

 mutual inductance, which is the property wanted, yet, if a 

 special value be given to it, any two coils in parallel will be 

 made equivalent to one. 



The condition required is obviously that Z, the generalized 

 resistance of the two coils in parallel, should reduce to the 

 form R + Lp. Equation (30c) gives Z ; to make the reduction 

 possible, on dividing the denominator into the numerator, the 

 second remainder must vanish. Performing this work, we 

 find 



h— — ■ f- j— —. ^— pj . > . . (Id) 



ri+r 2 l i -\-l 2 — 2m 1 K J 



which shows the effective resistance and inductance of the 

 coils in parallel, r x and r 2 being their resistances, and l lf l 2 , m 

 the inductances ; subject to 



n = ^ ; (Sd) 



r 2 l 2 ^-m ' 



giving a special value to m. which, if it be possible, will allow 

 the coils to behave as one coil, so that, when put in one 

 side of the Christie, the self-induction balance can be made. 

 This equation (8d) is the expression of the -making of coils 1 

 and 2 similar, in the extended sense, being the simple case to 

 which I referred. Let a unit current flow in the circuit of 



* Journal S. T. E. vol. vi. p. 303, « On Electromagnets." 



F2 



