FIGUKE AND STABILITY OF A LIQUID SATELLITE. 
179 
But if R is the distance between the point v 0 , //, <f> f on the ellipsoid and the 
external point v, /x, <J), this potential is 1/R. 
If we imagine a particle of mass Af/A. situated at v, /x, (f>, the above expression 
multiplied by M/k is the potential of the particle at the point v 0 , /x', <f>' on the 
ellipsoid. 
We have no need for the general expression for the potential of a particle situated 
anywhere in space at the surface of the ellipsoid, because it is only necessary to 
consider the case where the particle lies on the prolongation of the longest axis of the 
ellipsoid. In this case 
ry* 
[x= 1, (f) = fir, v = T , 
where r is the distance of the particle from the centre of the ellipsoid. 
But it is now no longer necessary to retain the accents to y!, <//, since they are 
only the co-ordinates of a point on the ellipsoid. 
Thus the potential of M/k, lying on the longest axis of the ellipsoid at a distance r 
from the centre, at the point v 0 , fx , <£ on the ellipsoid, is 
c/(f) 
f[WM €t(<l>)Jpda- 
For the types of functions denoted in “Harmonics” EES, OOS, OES, EOS, 
P/ (1) = 0, and for EOC, OOC, (£/ (i^) = 0. The only types for which f)/ (1) (£,* (|-7r) 
does not vanish are EEC, OEC; that is to say, cosine-functions of even rank. 
Accordingly the functions left are for i and s even, and P/C/ for i odd and s 
even; we may however continue to allow C/ to stand for both types. 
For brevity write 
V = few M ®<‘ WJp <!<?■ 
Thence, since (v 0 ) = 1, the potential may be written 
M 
kk 
m r) + f ^r 1 ) W (1) C/ ($7r) % s (v 0 ) W (/*) (fi* (<f>) |> for all even values of 5. 
k 
It must be observed that p/ and C/ occur as squares in ; they also occur 
twice in the numerator in the forms $/(l) %* (ji) and (& for) C/ (<j>). 
Again is of dimensions —1 in :}p/, and therefore €1;y j ) P/ ( v o) is of zero 
dimensions. From these considerations it follows that (/x) and (1/ ((f)) may be 
multiplied by any factors without changing the result, and further that may 
differ in its mode of definition from (/x) without producing any change. 
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