[ 35 ] 



II. Electromagnetic Integrals. 

 By Sir G. GREENHILL, F.R.S. 



(Received 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 (I., 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 commeicial use. 



The experiments have been carried out with the ampere balance invented by 

 VIEIAMU JONES, also with the Lorenz apparatus for measuring resistance (' Phil. 

 Mag.,' 1889, ' Phil. Trans.,' 1891, 1913), constructed at the charge of Sir ANIJHKW 

 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. MINCHIN, 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 



(1) M= TrAacose-, PQ 3 = A 2 + 2Aacos0 + a 3 + & 2 , AOQ = d, 



o 



and M is expressible by the complete E.I., I. and II. 



VOL. CCXX. - A 572. G [Published December 15, 1919. 



