238 Mr. T. J. FA. Bromwich on the 



where p is the surface-density, a is the radius of the disk, the 

 origin is the centre of the disk and its plane is the plane 

 of xy. Further, 



r 2 = ^ 2 -j-2/ 2 -f £ 2 , z = r cos 0, 



and P n stands for P n (cos#), Legendre's coefficient of order 

 n ; also c n is the coefficient of x n in the expansion of (l-f.r)^ 

 in powers of x, so that 



(l+ac)z=c + c 1 x+c 2 % 2 + c s w* + . . . . 



For the future we shall write cos = /jl, for brevity. 



At the plane of the disk (z = 0) it is easy to see that the 



two values of Y 1 are continuous, but that ?— 1 is discontinuous: 



a fact which agrees with what we know from the general 

 properties of the potential. But, apparently, at r — a V is 

 not equal to Y 1 ; and this is the point which I wish to clear 

 up, for, of course, there can be no discontinuity in the 

 potential and its differential coefficients at any point in free 

 space. From the previous results we have at r = a, 



V -Y 1 = 27r/C;«[(c 1 ~Co)+P 1 + fe~C 1 )P 2 + . . . 



+ (o"+i-€w)Pa.+ ...], (a*>0) 

 or 27rpa[(c 1 — c )— ?!+ (c 2 — Ci)P 2 + . ., 



+ (Cn + l-C»)P 2 ,+ • • •]• 0*<0) 



Hence, if there is to be no discontinuity, remembering that 

 P 1 = /a, we must have 



+/l=(c — cj + (<?,— c 2 )P 3 + . . . +{c n —c n+1 )F 2 >i+ • • ., (/a>0) 



— fJL = (c — Ci)+(C! — C 2 )P 2 -h • • • +(C„ — C„+i)P 2 ft+ . . . (i*<0) 



In order to test this, let us expand f(fju) in terms of 

 Legendre's coefficients, where 



/&»)=■+/. o*>o) 



and == -~A t (A t< ^)« 



We know that, with certain restrictions on the nature of 

 /(/a), of the same type as Dirichlet's conditions for Fourier's 

 series, we can write 



/M=|A B P»( At ), 

 »=o 



