552 
I’K0FJ<:SS01i G. H. DAPaVIX OX ELLIPSOIDAL HAPMOXIC AXALYSLS. 
liiiiitutioii the coefficients of tlie series assume simple forms (§ 8), and we liave thus 
definite, if approximate, expressions for all the functions which can occur in 
ellipsoidal analysis. 
In rio’orous expressions, and P/ are essentially different from one another, hut 
in approximate forms, when s is greater than a certain integer dependent on the 
degree of approximation, the two are the same thing in different shapes, except 
as to a constant factor. I liave, therefore, in § 9 determined up to squares of /3 the 
factors wherehy P® is convertible into ^0,®, and 0/ or S® into (£■ or S®. With the 
degree of approximation adopted thei'e is no factor for converting the P's when 
= 3, 2, 1. Similarly, down to s = 3 inclusive, the same factor serves for converting 
C/ into ff® and S/ into ^®. But for s = '2, 1,0 one form is needed for changing 
C into (Jf, and another for changing S into It may he well to note that there 
is no sine-function -when s is zero. 
The use of these factors does much to facilitate the laboilous reductions involved 
in the whole investigation. 
It is well known that the Q-functions are expressible in terms of the P-functions 
by means of a definite integral. Hence (1^® and Q* must have a second form, which 
can only differ from the other by a constant factor. Tlie factors connecting the two 
forms are determined in § 10. 
The second part of the paper is devoted to applications of the harmonic method. 
In ^ II the perpendicular from the centre on to the tangent plane to an ellipsoid 
and the area of an element of surface of the ellipsoid, are found in terms of the 
co-ordinates g, (f), and the constant ly. 
It is easy to form a function, continuous at the surface uhich shall ije a solid 
luirmoinc both for external and for internal space. Poissox’s equation then enables 
us to determine the surface density of whicli this continuous function is the potential, 
and it is found to he a surface harmonic ofy, ^ multiplied by the perpendicidar on to 
the tangent plane. This application of Poisson's equation involves the use of the 
(^-function in its integral form. Accordingly, if tlie serial form for the Q-function is 
adojited as a standard, the expression for the potential of a layer of surface density 
involves the use of the factor for conversion between the two forms of Q-function. 
Tills result may obviously he employed to determine the potential of an harmonic 
deformation of a solid ellipsoid. 
The potential of the solid ellipsoid itself may he found by the consideration that it 
is externally equal to that of a focaloid shell of the .same mass. It apjiears that in 
order to express the equivalent surface density in surface harmonics, it is only 
nece.s.sary to express the reciprocal of the square of the perpendicular on the tangent 
plane in that form. This result is attained by ex])ressing x~, y~, in surface 
harmonics. When this done, an application of the preceding theorem enables us to 
Avrite dowui the external potential of the solid ellip.sold at once. In § 12 the external 
potential of the solid ellij)soid is expressed rigorously in terms of solid harmonics ot 
degrees zero and 2. 
