46 REPORTS ON THE STATE OF SsCIzNcE.—1912. 
where R, Q, © are given by the Elliptic Integral of the First, Second, 
and Third Kind ; and a numerical result is obtained more quickly than 
by the use of the expansion in Spherical Harmonies. 
The expression of Q in terms of functions of the modulus c or 
¢, 18 
(59) Q = 2r(1 —f) — 2F zn fF’ — 2Kc'sn Af FP’, 
where 
(60). nafP—=", nofPeA—* anafroee® 
T2 2 T) 
and a Quadric Transformation is required, as in Maxwell's § 702, for the 
expression of 2 in the form given in (48), 
(61) Q = 27 (1—f) — 4K anf K’, 
and then 
(62) =) Bork 
rT) + T2 
M = 20QA = 4n(r, + 7) (K ~ H) = 8rv/(Aa) SS 
K 
where 
ie 6 Te -— 2v (Aa) 
ee tats. ee r+ 72’ 
Ks }(1+c)F, H = E ($7, «). 
When the disc is dished to a spherical surface of radius ¢, the 
potential V assumes a simpler form, given in the ‘American Journal of 
Mathematics,’ xxxiii., p. 387, by 
(64) V=H=oQ47'0', 
where ©’ denotes the apparent area at P’, the inverse point of P in 
the spherical surface, at a distance 7’ from the centre; and it is not 
easy to deduce the particular form in (58) by proceeding to ¢ = 09, to 
obtain a flat dise. 
In a more general form, the Elliptic Integral will appear in the 
standard shape 
1 dx 
(65) | (1, or x —h, or oy) Woe 
where X is a quartic function of z, which we suppose resolyed into 
factors 
(66) X = (e — a) (@ — B) (@— vy) (e@— 9). 
The reduction to a preceding form in the variable s is made by a 
linear transformation depending on the correspondence 
(67) x 5, *s, &  % con ok 
s Ga, "S))" Sai? Ssyeg 2; ae 
so that 
ga) 8 Ss Oe tir B tie Bs P= V8 = Oe 
z—-d o—S, xL— o— 58) L—-d o—S3 
