44 
SIR (4. GREENHILL ON ELECTROMAGNETIC INTEGRALS. 
reducing to ordinary conjugate function relations at a great distance from the axis 
O.r, where y is large. And with any conjugate system, x-^iy = f{u + iv), [dx, dy) 
are replaced by {dAi, d'v) ] thus for polar co-ordinates x + iy — du = drfr, 
dr = d^. 
* —. 
Tf the motion is not symmetrical about the axis Ox, and is not uniaxial, the S.F. 
does not exist; and in equation (4) an additional term is required for the variation 
with angle 0 of azimuth, so that in this general case 
(6) 
dx dx) ' dy \ dy > y d(fd 
= 0, 
expressing the resultant leakage or crowding-convergence of Q through an element of 
volume dx . dy/y d(j), when this is zero. 
]\I . 
Thus the result in (7), § 3, that ^— j- cos (/> is a P.F., —Q cos 0, is true for any S.F. M ; 
ijTrA 
for changing A into //, and putting M = V//, 
(7) 
_^/AcZM'> d /l fZM \ 
dx ^ y dx / dy^ y dy 
r/^V 
d^Y 
1 dY Y 
dx^ 
dy' 
// dy y' 
1 
.?/ 
jlx 
dx / 
dy^ -' dy 
so that V cos 0 satisfies equation (6) for 12. 
7. Maxwell shows further that 12 is the magnetic potential of any sheet bounded 
by the circle AB with uniform normal magnetization, so that, taking the plane circle 
AT), 12 is given by the normal component of tlie surface attraction of the circular disc 
AB, and so is the solid conical angle sulitended at P. 
lliis is true for any boundary AB ; for if dS denotes any elementary area enclosing 
a ])oint Q, the element of normal attraction, ^^^cosQl^M, is the element of surface 
4 A 
of unit sphere with centre at P, cut out of the cone on the Tiase dS. 
In Maxwells expression for P, surface potential of a spherical segment on tlie 
circular base AB, given in the form of a series of zonal harmonics (E. and M., § 694), 
he proves that 
(1) --^(Pr) = 0 
but he does not notice that 
(2) P = 12c-Mr^ = ldc + 12V, 
where 12^ is the solid ancrle 
in the sphere. 
or apparent area of the circle AB from the inverse point 
