SIR G. GREENHILL ON ELECTROMAGNETIC INTEGRALS. 39 



2. An integration of M with respect to b will lead immediately to V. JONES'S 

 expression for the mutual induction, L, between the circular current PP' on the 

 diameter 2A, and a uniform current sheet flowing round the cylinder on the diameter 

 2a, stretching from the circle AB a length b, up to the circle PP' ; drawing out the 

 circle AQB axially, like a concertina. 



The current sheet is taken as the equivalent of the close helical winding in the 

 ampere balance of the wire on the cylinder carrying the electric current and forming 

 a solenoidal magnet, of which a constant L gives a line of magnetic force, the one 

 passing through P, these lines circulating through the solenoidal tube and closing 

 again outside. 



In the hydrodynamical analogue L would be the stream function (S.F 1 .) of liquid 

 circulating through the tube. 



Employing the lemma of the integral calculus, for the line potential of MP at Q, 



'b . _ f i_i b u-1 b , _i PQ 



= r M db = r r* 



Jo Jo Jo 



and integrating by parts, with the lemma of the differential calculus 



/o\ d ,1 _i b _ A sin 8 b 



d0 i] PQ = "' 



/.\ T . / a ,,_, b V" f A a A sin bdO 



L = 27 rA (dn th ^ - J 27 rAa sm -^- - -^ 



the * marking a term which vanishes at the limits, and with 

 (5) 4AV sin 2 6 = 4AV- (MQ 3 -a 2 - A 2 ) 2 



(6) 



( a *-A a ) 2 ']bdo 



*-A a ) 2 '] 



MCpj 



introducing the complete elliptic integral, I., II., III.; and this is the expression 

 employed by V. JONES, but obtained by a complicated dissection. 



