74 
MR. J. H. JEANS ON THE EQUILIBRIUM 
The solution for ^ will be a function of -q and c. The maximum values for [ -q \ and 
— will therefore be functions of c, and will be expressible as series of ascending powers 
of c. Substituting these values respectively in {rj) and Q (p), we obtain two series 
in ascending jDOwers of c. Let the radii of convergence of these two series be and Co 
respectively. Then (p (rj) is convergent inside and over the boundary jDrovided c < C| 
and xjj (t]) is convergent outside and over the boundary provided c < c^. The solution 
for ^ found by expansion in jDowers of c wiU satisfy the conditions which it is assumed 
to satisfy in § 3 so long as c is less than the smaller of the two quantities Cj and Co. 
§ 9. Let us supj)ose our solution in powers of c to have been obtained. The values 
of V,' and Vq can be regarded as power series of the variables £ or p, and as such wHl 
liave circles of convergence in space, having the origin as centre. Let V,- be 
convergent inside a circle of radius R^, and Vg outside a circle of radius R^. When 
c = 0, R^ < Ri; the circle Rj being wholly inside, and the circle R^^ being wholly 
outside the surface S. When c reaches the value Co, the circle Ro touches the surface 
S ; and for values of c greater than Co, the circle Ro either intersects, or lies outside, 
the surface S. 
Suppose that Cjj < : then for values of c such that 
Co C Cj 
the circle R^ lies wholly outside S, while the circle R., does not lie wholly inside S. 
Thus if V/ and Vg are defined by the same power series which have been found for 
them for values of c less than Co, V; will be convergent at all points inside S, but Vg 
will not l)e convergent at all points outside S. We have now to inquire whether it 
is possible for V, to give the true value of the internal potential, even when the series 
found for Vg fails to represent the external jDotential. 
Let the equation to the surface be written in the form 
f {trj, c)— 0 .(33), 
and let the solution be v’ritten in the form 
^ z= c) -ir xjj {q, c) .(34). 
We shall only consider the case in which c < c^, so that oui' former function (p (q) is, 
by hvpothesis, convergent at the boundary and at all points inside it. We take 
(p {q, c) to mean the same thing as our former (q), and siq^pose xjj {q, c) defined by 
equation (34). Now the value of ^ cannot become infinite at any jioint on the 
boundary, since we suppose the bounding surface not to extend to infinity for any 
value of c. Hence it follows tliat our function xp (q, c) will be finite at all points of 
the boundary so long as c < Moreover, in the region in which our formei' xp (q) is 
convergent {i.e., in the region r > R^), ijj {q, c) must become identical with xp (q), and 
will therefore be finite. The function 
xp {q, c) + xp (^, c) 
w 
ill therefore be finite at the surface S, and will vanish at infinity to an order at 
