44 SIR G. GREENHILL ON ELECTROMAGNETIC INTEGRALS. 



reducing to ordinary conjugate function relations at a great distance from the axis 

 Ox, where y is large. And with any coiijugate system, x + iy = f(u + iv), (dx, dy) 

 are replaced by (du, dv) ; thus for polar co-ordinates x + iy e logr+w ( du = drfr. 

 dp = d. 



If 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 <j> of azimuth, so that in this general case 



d I dU\ d / dti\ . d\2 



(6) T~(V-j-} + j-(y-J-) + ~JTa- () ' 



dx \ dx I dy \ dy ] y d<(, 



expressing the resultant leakage or crowding-convergence of Q through an element of 

 volume dx . dyjy d<j>, when this is zero. 



M 



Thus the result in (7), 3, that - r- cos </> is a P.F., Q cos 0, is true for any S.F. M ; 



iTTJ\. 







for changing A into //, and putting M = V//, 



l 



dx \y d.r ! d// \y dy ' 



= d 2 V d?V 1 dV _ V 

 dx 2 d>/ 2 ;, dy y 2 



so that V cos </> satisfies equation (6) for 12. 



7. MAXWELL shows further that i2 is the magnetic potential of any sheet bounded 

 by the circle AB with uniform normal magnetization, so that, taking the plane circle 

 AB, 12 is given by the normal component of the surface attraction of the circular disc 

 A I), and so is the solid conical angle subtended at P. 



Tins is true for any boundary AB ; for if dS denotes any elementary area enclosing 



a point Q, the element of normal attraction, ^-jCosQPM, is the element of surface 



I (.^ 



o 



f unit sphere with centre at P, cut out of the cone on the base dS. 



In MAXWELL'S expression for P, surface potential of a spherical segment on the 

 circular base AB, given in the form of a series of zonal harmonics (E. and M., 694), 

 he proves that 



but he does not notice that 



r 



where Q' is the solid angle or apparent area of the circle AB from the inverse point 

 jn the sphere, 



