PROFESSOIL G. IT. DARWIN ON ELTJPSOIDAL HARMONIC ANALYSIS. 
that follows would then be ei^ually applicable ; but, in order not to complicate the 
statement by continual reference to alternative forms, I shall adhere to the first 
form as a standard. 
In this paper a closely })arallel notation is adopted for the ellipsoid of three 
unequal axes. The squares of semi-axes of the fundamental ellipsoid are taken to 
, —/d, 0, and tlie three co-ordinates are still v,/ jl, cf). Although their 
1 fO 
geometrical meanings are now by no means so simple, they may still be described a.s 
radial, latitudinal, and longitudinal co-ordinates. As before, we might equally well 
1 + /3 
start with a fundamental ellip.soid whose squares of semi-axes are /d 0, and 
replace v- liy {h where = — p'\ All possible ellipsoids are comprised in either of 
these types by making /S vary from zero to infinity. But it is shown in § 2 that, by 
a proper choice of type, all j^ossible ellipsoids are comprised in a range of ^ from zero 
to one-third. When /3 is zero we have the spheroids for which harmonic analysis 
already exists; and u'hen ^ = I the ellipsoid is such that the mean axis is 
the square root of mean square of the extreme axes. The harmonic analysis for this 
class of ellipsoid has not been yet Avorked out, liut the method of this paper Avould 
render it possible to do so. We may then regard /3 as essentially less than and 
may conveniently make de\’elopments in poAvers of /3. 
In .spheroidal analysis, for space internal to an ellipsoid tAA'o of the three func¬ 
tions are the same P-functions that occur in spherical analysis; one P being a 
hmctlon of v, the other of /x. The third function is a cosine or sine of a nndtiple of 
the longitude 6. In external space the P-function of p is replaced by a Q-function, 
being a solution of the differential ecpiation of the second kind. 
The like is true in ellipsoidal analysis, and Ave have P- and Q-functions of p for 
internal and external space, a P-function of p, and a cosine- or sine-function of cj). I 
Avill noAv for a time set aside the Q-functions and consider them later. 
There are eight cases to consider (§ 4) ; these are determined by tlie eA'enness or 
oddness of the degree i and of the order 5 of the harmonic, and by the alternatiA’e of 
Avhether they correspond Avith a cosine- or sine-function of </>. I indicate these eight 
tyjAes by the initials E, O, C, or S —for example, EOS means the type in Avhich i is 
even, s is odd, and that there is association AAotli a sine-function. 
It appears that the neAv P-functions fall into tAvo forms. The first form, Avhich I 
Avrite Pf, is found to be expressible in a finite series in terms of the P/ AA'here tlie 
P’s are the ordinary fimctioiis of s})herical analysis. The terms in tliis series are 
arranged in poAvers of /3, so that the coefficient of P*±-''' has as part of its 
/ ** ^2 _ 
vr t +3 P/ (a) or 
a/ i‘+^ ^^2 P/(/a) is expressible l)y a series of the same kind as that for Pf. 
^ 1 -^ 
Amongst the eight types four iiiA’olve P-functions and four P-functions ; and if for 
