SIR G. GREENHILL ON ELECTROMAGNETIC INTEGRALS. 55 



end circular plates. In this way we have arrived at the expression for W, the 

 surface potential of the circle on AB, in the form 



(2) W = Pa-QA-06, 



, Q x (dW dW .dW\ p 



' ~ ' ~db) = P> ~ Q > - 



in accordance with EULER'S theorem for a homogeneous function, in this case W is 

 a homogeneous function of the first degree in the three variables a, A, b. 



At a meeting of the London Mathematical Society, November 11, 181)9 (' Proc. 

 L.M.S.,' vol. Ill, p. 8), Prof. CAYLEY presiding, the Secretary, Mr. JENKINS, read 

 a letter from Mr. CLERK MAXWELL asking the following question: "Can the 

 potential of a uniform circular disc at any point be expressed by means of elliptic 

 integrals ? I am writing out the theory of circular currents in which such quantities 

 occur." 



But the result is obvious from the theorem above of a homogeneous function, 

 so that 



fA\ w t 



(4) W = -j- a+ -j A+ 



-j -jj-, 

 da dA. db 



in which 



/,\ dW Cade dW i+acoBdde dW 



(5) -y = P = -^- , -r = - = -Q, and =r = Q, 



da J PQ dA PQ db 



for any shape of the disc. 



Prof. CAYLEY'S attention was thereby directed to the subject, and he extended the 

 investigation to the elliptic disc (' L. M.S.,' vol. VI). 



15. Integrate P with respect to b to obtain the skin P.F. of the curved surface 

 of the cylinder, drawing out the circle on AB like a concertina, 



(i) 



suppose, an intractable integral, th~' (6/PQ) being the potential of the generating 

 line element of length b. 



But cos 0th" 1 :p7=r d0, as in the expression for L in 2 (2), is tractable and given 

 J " y . 



in finite terms, while sin 6 th" 1 =-=r dO is non-elliptic, expressed in the variable cos Q. 



J s: y 



The integral I cannot be made to depend on a finite number of elliptic integrals, 

 but requires to be expanded in an infinite series, and so we say it cannot be expressed 

 in finite terms. 



