rUOFESSOK G. II. l)Ai;\MX ox ELLIPSOIDAL IIAIEMOXIC AXAIASIS. 
4(i.3 
Now let /3 increase from 0 to cc . 
As ^ passes from 0*to •§-, form (l) is appropriate. 
As passes from ^ to I, fiy decreases from -j to 0, so that form (2) is appropriate. 
Lastly, as /? from 1 to oc , increases from 0 to so tliat form (3) is 
appropriate. 
But we might ecpially well have written forms (1) and (3) so as to involve and 
form (2) so as to involve v, and it follows that all possible ellipsoids are comprised in 
the range of /3 from 0 to provided that the type be appropriately chosen. 
The developments in this paper are made in powers of /3. It will, therefore, be 
well to show that there is a class of ellipsoids, analogous to ellipsoids of revolution, 
which might form the basis of developments similar to those carried out below. 
Ellipsoids of revolution are defined by the condition 
rr — = fr — c\ or O' = />'. 
In the class to which I refer 
a- - c- = C- - Ir, or C- = l + /d). 
Ellipsoids of this kind are given by /3 — = T : for in this case 
h- = Tfrr -fi *"')• They are also given hy 
/3 = X , — = /3,, = I; for then c- = + h~). 
Hence if we only allow ^ to range from 0 to yS = 0 corresponds with ellipsoids 
of revolution, to which spheroidal harmonic analysis is applicable ; and /3 = ^ 
corresponds with this new class for which the corresponding analysis has not yet 
been worked out. 
AVe shall see below that the solid harmonic for this case where /3 = L will be of 
the form B {v) B (p) E {(b), vrhere B and E satisfy the equations 
(.,2 4- l)(v2 - l)ft + - l(i + + »'B = 0, 
■ ' (hr dv 
cos 2fp ; — sin 2(i — + i (i fi- 1) E cos 2(b — .s'-E = 0. 
d(p^ ' d(j^ 
I am not clear whether or jiot it would be advisable to proceed ah initio from tliese 
equations, but at any rate I shall show hereafter how the B- and E-functions may be 
determined from the analysis of the present paper with any degree of accuracy 
desirable. 
If it were proposed to use the functions corresi^onding to /S = as a basis for the 
development of general elli])Soidal harmonics, we should have to assume 
VOL. cxcvri. 
3 o 
