Arc Oscillations in Coupled Circuits. 661 



the experiments there described the circuits were so adjusted 

 that the frequency of one of the oscillations corresponded 

 either to one of the harmonics of the other, or to the perfect 

 fifth above it. It was pointed out that in the latter case 

 it was necessary that the two notes of the system should 

 be equally stable in order that the double oscillation curve 

 might be produced, and that the note then heard was an 

 octave below the lower of the two primaries, being in fact 

 their difference tone. 



It was thought desirable to continue these experiments, 

 using some of the smaller intervals, in order to find out 

 whether the same conditions hold, whether the difference 

 tone is produced, and whether the same method of calculating 

 the frequencies of the two oscillations also applies to these 

 cases. 



The smaller ratios - are obtained by diminishing the co- 

 efficient of coupling of the two coils, and if we assume as the 

 approximate condition of equal stability of the two notes 

 L 1 C] = L 2 C 2 ' X ', the value of this coefficient may be calculated 

 for any given ratio of the two frequencies. 



If, with the usual notation for the constants of the two 

 circuits, we put 



1/LA:=]SY, 1/L 2 C 2 = X 2 2 , WjL.L^P, 



the equation for the two frequencies, n v ?i 2 , becomes 



87rV = f4p [ N i 2 + N 2 2 ± ^{(N^-N^ + ^PNAV}]. 



Assuming the condition N!=N 2 , and writing m for the 

 ratio of the frequencies, this leads to 



7 _ m 2 -l 



■ s+r 



Taking as an example m = 3/2, as in the former experiments, 

 this gives /; 2 = *1479. The experimentally determined value 

 of & 2 for the two coils was "1483. During the singing-arc 

 experiments the value of Jc 2 would be rather less than this, 

 owing to the existence of self-inductance in the arc. In 

 order to obtain the ratio ?n = 4/3 the value of k 2 should be, 

 according to the above formula, '0784. 



There is no doubt, however, that the above condition for 

 equal stability, Nj = N 2 , 1S on ^J approximate; the value of 

 the secondary capacity which makes the two notes equally 

 stable depends also to some extent upon the mutual inductance 

 of the coils. 



* Qf. E. T. Jones, Phil. Mag. January 100?. p, 41. 



