[ 35 ] 
ir. Electromagnetic Integrals. 
By Sir G. Greenhill, F.R.S. 
(Eeceived April 22,—Read May 9, 1918.) 
These are the integrals, elliptic integrals (E.I.) for the most part, and of the first, 
second and third kind (L, II., III., E.I.) arising in the practical problem of the 
measurement and determination of the electrical units, for their regulation and 
definition by Act of Parliament in commercial use. 
The experiments have been carried out with the ampere balance invented by 
ViRiAMU Jones, also with the Lorenz apparatus for measuring resistance (‘Phil. 
Mag.,’ 1889, ‘ Phil. Trans.,’ 1891, 1913), constructed at the charge of Sir Andrew 
Noble and the British Association, in use at the National Physical Laboratory 
(N.P.L.), Teddington. 
A description of this current weigher, ampere balance, is given by Ayrton, 
Mather and F. E. Smith in the ‘ Phil. Trans.,’ A, vol. 207, 1907, and the theory is 
developed, with a description of the accuracy of measurement obtainable, to be 
recorded in the Act of Parliament. Also of the Lorenz apparatus, by F. E. Smith, 
in ‘Phil. Trans.,’ 1913. 
Our object here is to revise and simplify the mathematical treatment required in 
these calculations and to present the theory in a form adapted for elementary 
instruction ; at the same time, to reconcile the notation and results of the various 
writers, ViRiAMU Jones, G. M. Minghin, and others, and to standardise them in 
accordance with Maxwell’s ‘ Electricity and Magnetism ’ (E. and M.). 
1. Starting then on § 703, E. and M., it can be shown that all the results required 
„ can be made to originate and grow out of Maxwell’s expression for M, mutual 
induction of two parallel circular currents on the same axis, in circles of radius 
OB = a and NP = A, a distance ON = b apart, as on fig. 1 and 2 (p. 38), and M is 
given in his notation by 
(l) M = [ 27rAacos0:^, PQ^ = A^+2Aa cos 0 + + AOQ = 0, 
Jo PO 
and M is expressible by the complete E.I., I. and 11. 
VOL. CCXX.-A 572. G [Published December 15, 1919. 
