ol4 PROFES.S(Jl; (;. II. DAKWIX ox ELLIPSOIDAL IIAPMOXIC AXALVSIS. 
Ill tliis we may substitute the expressieus for if-, :• found above. 
It may be Avortb iiotiug that 
Qi^hi') . , (%(r) __1_ 
Pfv) Pp;.) - - \Hy ■ 
Also p,‘(>’)Qi‘o + i3,*(oQi‘(>') + r,(OQiO + ro(>'iQ„(o = <1. 
foll.ovs fvoin the fact that if a, h, e are the axes of the ellipsoid, and 
A~p(’ (^o) )■ homogeneous 
- 0 *^Lt 
function of degree — I in «, Ij, c, and therefore 
t/'I^ , , '/'P . </'I^ 
u - + /v— + c — = — H' . 
< 1 ,< 
ilh 
lie 
§ 13. Prcpai'cition fo)‘ the Integration of the square of' a surface harmonic 
over the EUipsoiel. 
If it is intended to express any function in harmonics, it is necessary to know the 
integrals oA'er tlie surface of the ellipsoid of the squares of surface harmonics 
multiplied by the ])er])endicular on the tangent plane. 
'I’he surface'harmonic has one of tlie eight forms 
(g) or 
P-{g}] X 
€f{ch} 
or 
C ;(</>! 
S/((/))_ 
and the P-functions are expressible in terms of the Ps where 
(1 - fp I'dfo 
I.il V7y/ 
if - \y. 
1 shall in this portion of the investigation frequently write g = sin 0, and shall 
omit the p or 9 or (f in the P-, C-, S-functions. Also I may very generally omit the 
subscri])t g as elsewhere. 
If da denotes the element of surface of the ellijisoid, and 
M = Bv{v^ 
t + /3 i - 
1 - /d/ ’ 
BO that 3 IT M is the \'olume of the ellipsoid, Ave have, by (5U) of § IP 
rda 
JI(l-/3)> (cos^ 0 + rg ~ 
dOiip) (1 — /3cos 20)^ (j Af — mi- ^)- 
I'heu 
I l>(Ai')Oo- = M(1 - /3)q| — 
COS= d + g, - “-Hg 
/3cos-2(py-{\f^^ - sin-P) 
where the limits of 6 are ivr to — lir, and of </> are l’tt to 0. 
