of Edinburgh, Session 1870 - 71 . 
503 
which is the expression given by Maxwell (London Math. Soc. 
Proc ., Yol. III., No. 34, 1871). Although, for simplicity, P has 
here been supposed a scalar, it is obvious that in the result above 
it may at once he written as a quaternion. 
I 
4. On an Expression for the Potential of a Surface-distribution, and 
on the Operator TV = </(**)' + (f) + (|)* • 
If p he the vector of the element ds , where the surface density 
is /p, the potential at a~ is 
ffdsfp FT(p - er) , 
F being the potential function, which may have any form whatever. 
By the preceding Note this may be transformed into 
ffdsfp e S<rV FTp; 
or, far more conveniently for the integration, into 
ffdsfp Viv, 
where A depends on the constituents of <r~ in the same manner as 
V depends on those of p. 
A still farther simplification may be introduced by using a 
vector <r^, which is finally to he made zero, along with its corre¬ 
sponding operator A 0 , for the above expression then becomes 
rr 7 &p(A — A o) TT'rp 
ffdsz fai FIV, 
where p appears in a comparatively manageable form. This is the 
expression to which the title of the Note refers. It is obvious 
that, so far, our formulae are applicable to any distribution. We 
now restrict them to a superficial one. 
Integration of this last form can always be easily effected in the 
case of a surface of revolution, the origin being a point in the axis. 
For the expression, so far as the integration is concerned, can in 
that case be exhibited as a single integral 
