ON FARADAY'S LINES OF FORCE. 223 



Within the shell <u cannot become infinite ; therefore a> = C 1 is the solution, 

 and outside a must vanish at an infinite distance, so that 



..^ " "" V " '' 



is the solution outside. The magnetic quantity within the shell is found by last 

 article to be 



o T n 3 _ ^& dy, 



* 6a 2k+k' dz dy 



therefore within the sphere 



7.n 1 



Q) = - - . 



Outside the sphere we must determine <a so as to coincide at the surface 

 with the internal value. The external value is therefore 



= _ 

 ~ 



__ 

 3k+k' r 



where the shell containing the currents is made up of n coils of wire, con- 

 ducting a current of total quantity 7 r 



Let another wire be coiled round the shell according to the same law, and 

 let the total number of coils be n' ; then the total electro-tonic intensity 

 round the second coil is found by integrating 



EI t = I <ua sin 6ds, 

 Jo 



along the whole length of the wire. The equation of the wire is 



ntr 

 where n' is a large number; and therefore 



ds = a sin 0d<f>, 



r, T 4ir , , 2ir , T 

 .'. EI t = wa'n = -annl 



E may be called the electro-tonic coefficient for the particular wire. 



