and the Theory of the Induction-balance. 127 



as completely as possible without taking into account the elec- 

 trostatic capacity of the wires. It would be very interesting 

 to obtain the perfectly general equations ; but they are very 

 complex, and appear to be beyond me at present. 



General Theory of Current-induction. 



4. The general theory of the establishment of a current in a 

 number of circuits of known resistance, in one or all of which 

 a given electromotive force begins to act, was begun by Helm- 

 holtz and Thomson, who independently showed that the fact 

 of the existence of induced currents, discovered experimentally 

 by Faraday, as well as the laws of the same, could be deduced 

 mathematically from the electromagnetic attractions of Oersted 

 and Ampere by means of the principle of the conservation of 

 energy. The complete theory, it is needless to say, is to be 

 found in Maxwell's ' Electricity,' and also in that admirable 

 digest of the whole subject, Professor Chrystal's article in the 

 Encyclopaedia Britannica. 



Suppose there are n circuits of resistances r lf r 2 , . . . , in each 

 of which " impressed " electromotive forces E 1? E 2 ... begin 

 to act at the same instant ; then the strengths of the currents 

 * 1? i 2 . . . in the several circuits at any time t afterwards are to 

 be found by the following set of n equations — 



Ei-^'i= j t (Mnh + M 12 ? 8 + . . . + M ln i n ), " 



E 2 -r 2 2 2 =^(M 21 ^ + M 22 ? 2 + ... + M 2 ^ n ), ^ ^ - ,y 



E n - r n i n = | (M n A + M n2 i 2 + ... + M nB i n ), ^ 



where M^ is a double integral, called the coefficient of mutual 

 induction between the circuit I and the circuit k. 



Induction-balance. 

 5. Now in the induction-balance as ordinarily arranged 

 there are three such circuits, with the coefficients M all con- 

 stant, and two of them without any electromotive force. The 

 first, or battery-circuit, consists of a battery and two equal 

 coils, and it also contains some kind of intermittent break or 

 microphone ; the second is the telephone-circuit, and consists 

 of two equal coils and a telephone ; and the third is complete 

 within the coin or other piece of conducting matter in the 

 neighbourhood of one pair of the coils of the other circuits. 

 The general value of the telephonic current at any instant 

 after a make has occurred in the primary would therefore be 



