98 
MR. J. IT. JEANS ON THE EQUIITBRTT’M 
In this same figure the thick curves represent the locus, 
cp (r, e) = 0, 
calculated upon the supposition that the calculated terms ol {r, 9) give a 
sufiiciently good appi'oximation to the whole. For the greater part of the curve 
this assumption is not justifiable, so that tlie curve recpiires adjustment, the 
amount of this adjustment increasing as we recede from tlie shaded portions of the 
plane. The most important points on the curve are those at which ddjdr — 0. 
Tliese may with sufiicient accuracy for our present purpose be taken to be r = 2, 
9—1, and r = — 2, 9 = — I. 
§ 28. The points at which the axis (f) = 0 meets the curve of which the erpiation is 
(128) are given by 
cp [r, 9) = 0. 
Fig. 3 accordingly enables us to trace the motion of these points as we move along 
the linear series, i.e., as 9 increases from zero upwards. At ^ = 0 we have, of course, 
two equal and opposite roots —r = rh .^/5. As 9 increases the positive root increases, 
wdiile the negative root numei'ically decreases. Remembering that the centre of 
gravity of the curve must remain at the origin, we see that this indicates a general 
thickening of the half of the curve in which (ji > 77 2, with a diminution in the 
thickness of the forward half, and consequent lengthening of this half These 
features become more marked as 9 increases, until we reach the value 9 = 1, at which 
a new feature presents itself For liere there are two new roots occurring at the 
point r = 2. This indicates that the fluid separates into two portions when the 
value 9=1 is reached, the point of separation being r = 2 (approximately). We 
are at once struck l)y the great inequality in size between the })riniary and satellite ; 
the former extending approximately from r = — 2 to r = + 2, and the latter only 
from r = 2 to r = 2^. The ratio of the linear dimensions Avill therefore be some¬ 
thing like 8 to 1, but it must be remembered that our results require considerable 
correction on account of the imperfections in oui- approximations. 
§ 29. In fig. 4 the thick curve is the elliptic cylinder corresponding to d = 0, and 
the dotted curve is the adjacent curve corresponding to a small value of 9 {9 = \). 
