PEOFESSOE G. PI. DAEWIX OX ELLIPSOIDAL IIAEMOXIC AXALYSIS. 
553 
Since y'^, have been found in surface harmonics, we can also write down a 
rotation-potential about auy one of the three axes in the same form. 
The internal potential of a solid ellipsoid does not lend itself well to elliptic 
co-ordinates, but expressions for it are given in § 12. 
If it be desired to express any arbitrary function of y, (f) in surface harmonics, it is 
necessary to know the integrals, over the surface of the ellipsoid, of the squares of 
the several surface harmonics, each multiplied by the perpendicular on to the tangent 
plane. The rest of the paper is devoted to the evaluation of these integrals. No 
attempt is made to carry the developments beyond /3'\ although the methods 
employed would render it possible to do so. 
When s is greater than unity, it appears that it is legitimate to develop the 
function to be integrated in powers of ^ ^ ; and when this is done, the integration, 
although laborious, does not present any great difficulty. 
But when s is either 1 or 0, the method of develoj^ment breaks down, because it 
would give rise to infinite elements in the integrals at the poles where is unity. 
However, portions of the integrals in these cases can still be found by the former 
method of development. As to the residues winch cannot he so treated, it appears 
that they depend on integrals of the forms 
cos-" 0 (W 
(1 —k ' sin- 0) 
\ and j COS'" 0 {1 — k" sin^ 0)^ (10, 
where k '^ is nearly equal to unity. 
Development of the square-roots in powers of k ' is useless on account of the slow 
convergence, and it is required to find series wliich proceed by powers of /c', w'here 
K' = i — K X 
By a somewhat difficult investigation, in respect to which I owe my special thanks 
to Mr. Hobson, the needed series are found (§ 19). 
It appears that portions of the two integrals involve logarithms which become 
infinite wlien k vanishes. Since, in the application of these integrals, the vanishing 
of K implies the vanishing of (3, we appear to be met by a difficulty. It is known 
that in spheroidal analysis no such terms a})pear, and we may feel confident that 
they cannot really exist in ellipsoidal analysis. In § 20 it is j^roved that the 
logarithmic terms do as a fact disappear. The residues of the integrals in the cases 
.s = I, 0 are thus found, and added to the previous portions to form the complete 
results. 
The second part of the j^aper ends (§ 22) with a list of tlie integrals of the squares 
(fi’ the surface harmonics for all values of s, as far as the squares of /3. 
Finally, an appendix below contains a table of all the functions as far as i — 5, 
5=5. It is probable tliat for the higher values of s the results would only 1)e 
applicable wlien is very small. 
VOL. CXCVII.~A. 4 B 
