RESISTANCE BY A METHOD BASED ON THAT OF LORENZ. 
71 
of the disc nearer to it, and the second part gives the mutual inductance M 2 of the 
coil and the circumference of the other disc. The difference of these is required, so 
that M = M x — M 2 . 
To find M t , two mutual inductances were calculated, viz., that between D x and a 
coil of length BC, and that between D 2 and a coil of length AC. The difference of 
these mutual inductances gives M x . If the coils on the four cylinders are exactly 
similar the values of are identical. 
To find M 2 for coils Nos. 1 and 4, the mutual inductances were calculated between 
D 2 and a coil of length BK and that between D 2 and a coil of length AK. The 
difference is equal to M 2 . To find M 2 for coils Nos. 2 and 3, the mutual inductances 
required are that between D 2 and a coil of length EK and that between D 2 and a coil 
of length FK. 
In the Lorenz apparatus, the distances of the coils from the discs can be varied. 
This changes the mutual inductance and the rate of change of M with variation of 
axial distance must therefore be known. 
To find M a we used the following formula due to J. Viriamu Jones # 
m = e(A+«)a jLp? + £ (F-n)}. 
In this expression, 0 is the angular length of the helix, A the radius of the helix, 
a the radius of disc or contact circle, and x the axial length of helix. 
c 2 = 4A«/(A + a) 2 , c' 2 = 1 — c 2 , 
k 2 = iAa/(A + a) 2 + x 2 , k' 2 = 1 —k 2 , 
F, E, and II, are complete elliptic integrals of the first, second, and third kinds 
respectively ; F and E are to modulus k, and 
n = r dyfr 
Jo (l—c 2 sin 2 ^) (l—Fsin 2 ^) 1/2 
Putting c'/F = sin 6, the quantity (F —II) can be expressed in terms of complete 
and incomplete integrals of the first and second kindst ; thus 
c~ l k' 2 sin /3 cos £ (F—II) = — |tt— F (k) F (F, /3) + E (k) F (F, 0) + F (h) E (F, /3). 
The various elliptic integrals required were calculated by interpolation from 
Legendre’s tables, but as a check on possible misprints in the tables a number of the 
integrals were calculated directly by successive quadric transformation 4 
The dimensions chosen for calculating the values of M x are given in Table IX. The 
* J. Y. Jones, ‘Roy. Soc. Proc.,’ vol. 63, p. 198, 1898. 
t Cayley, ‘ Elliptic Functions,’ § 183. 
t Cayley, Chapter XIII. 
