Variable Mutual Inductances. 161 
the self inductance of the arm P being L, (the secondary 
coils of M) and that of Q being L 2 . Let H be a source of 
periodic current and Gr a vibration galvanometer tuned to 
resonance with it, so that Ave may take the wave form of the 
Fisr. 6. 
currents to be a sine curve. Let the instantaneous potentials 
of the three upper corners be r l5 0, and v 2 respectively, and 
the instantaneous values of the currents into the upper corner 
be i 13 i 2 , and i as marked. Let p = 27rn where n is the 
frequency, and for convenience of writing let p^ — 1 be 
denoted by a, so that a 2 = — p 2 . The mutual inductance M 
may be made positive or negative according to the way in 
which the coils are connected ; and in all that follows we 
might write + M for M throughout. When the galvanometer 
shows a balance, v 1 =v 2} and the instantaneous value of the 
current through G is zero. 
Also . . . 
i= — i 1 — i 2 . 
Accordingly we may write 
(P + L^)**! — Mo2=(Q + L 2 «)i 2 ; 
thpretore 
[P + (L 1 + M) a ]/ 1 =[Q+rL 2 -M>]i :i 
also R« 1 =St 2 . 
Hence 
S[P+(L 1 +M>]=R[Q-f(L 2 -M>]. 
Equating the real and imaginary parts each to zero we have 
SP = QR, (1) 
and S(L 1 + M)=R(L 2 -M) *(2) 
* (Nov. 22, 1907) I find that a case slightly more complicated than 
that of Equation (2) has been worked ont and applied in an ingenious 
manner bv Gratz (Wied. Ann. vol. 50. p. 766, 1893). 
Phil Mag. S. 6. Vol. 15. No. 85. Jan. 1908. M 
