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 = Ba—QA—115, 
(3) 
(<m dw dw\ r, ^ .. 
\ da' dA’ dh) ^ ’ 
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, h. 
At a meeting of the London Mathematical Society, November 11, 18G9 (‘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 
( 4 ) 
dW dW , dW . 
W = a+ -TV- Ah— rr- 5, 
da 
dA 
dh 
in which 
dW T, {add dW f+«cos0 d0 r\ j p, 
^ = P = JpQ’ = J FQ " = -Q’ 
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 h to obtain the skin P.F. of the curved surface 
of the cylinder, drawing out the circle on AB like a concertina, 
( 1 ) 
Pd5 = 
C 
a do dh 
~P^ 
I, 
suppose, an intractable integral, th ^ (5/PQ) being the potential of the generating 
line element of length h. 
cos 0 th"^ pQ do, as in the expression for 
in finite terms, while j sin 0 th ^ dO is non-elliptic, expressed in the variable cos 0. 
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. 
