PROFESSOR G. H. DARWIN ON ELLIPSOIDAL HAR^IONIC ANALYSIS. 54‘) 
Type EOS U + 8/3(./ + 2 )+si 4 ^'(./H^Q; —96) 
Type OOC | p ( P/(/a) )~ da 
)zf:' ’ (3i eio)+ Fb^-(29/H mi+ssi) ( 
Type EOC f p(PHriCmY<f<^=0i^p+Mj+^)+sh0m'+7ii+i») 
Lastly, when .s - 0 ; 2 = - ^ (^■+1) = -/. There are only two types— 
Type EEC fp(P,(ia) €i{<p))\da = ^^J~^[l+^/Bj + -h^^^ 
Type OEC f/> {\Mp)Oi (<^))' da- = ^^ [l+MZ+ow/^' (7/-2y-24l. 
PAPvT III. 
Summary. 
The symmetrical form in which Lame presented the three functions Y’hose product 
is a solid ellipsoidal harmonic is such as to render purely analytical investigations 
Loth elegant and convenient. But it seemed to me that facility for computation 
might he gained l)y the surrender of symmetry, and I have acted on this idea in the 
preceding paper. 
Spheroidal analysis has been successfully employed where the ellipsoid is one of 
revolution, and it therefore seemed advisable to make tliat method the point c)f 
departure for the treatment of ellipsoids with three une(|ual axes. In splieroidal 
harmonics we start with a fundamental prolate ellipsoid of revolution, with imaginary 
semi-axes — ky/—l, 0. The position of a point is then defined by three co-ordi¬ 
nates ; tlie first of these, v, is such that its reciprocal is the eccentricity of a meridional 
section of an ellipsoid confocal with the fundamental ellipsoid and passing through 
the point. Since that eccentricity diminishes as we recede from the origin, i> plays 
the part of a reciprocal to the radius vector. The second co-ordinate, /r, is the cosine 
of the auxiliary angle in the meridional ellipse measured from the axis of symmetry. 
It therefore plays the part of sine of latitude. The third co-ordinate is simply the 
longitude <^. The three co-ordinates may then be described as the radial, latitudinal, 
and longitudinal co-ordinates. The parameter h defines the absolute scale on which 
the figure is drawn. 
It is equally possible to start with a fundamental oblate ellipsoid \vith real axes 
k, /.’, 0, We should then take the first co-ordinate, as such that = — r". All 
