76 ME. J. H. JEANS ON THE POTENTIAL OF ELLIPSOIDAL BODIES, AND 



some curve at present undetermined. Hence, if the present paper is sound, we 

 should anticipate that DARWIN obtained a linear series instead of a rectangle of 

 configurations by the introduction of some adventitious condition not necessary to 

 equilibrium. 



39. DARWIN supposes his pear defined as far as the second order by the equation 



T = c-eS 8 -2/' S/, ......... (163) 



i 



where T gives a measure of the surface displacement, e is a parameter analogous to 

 our e (although not numerically the same), and the quantities f* are coefficients which 

 must vary as we pass along the linear series, but are constants, as is also e, for any 

 single figure of equilibrium. The quantities S s , S," are ellipsoidal harmonics. DARWIN 

 supposes K to be a small quantity of the first order, and the coefficients f' to be small 

 quantities of the second order. The energy E of the distorted ellipsoid will differ 

 from that of the original ellipsoid from which the displacement r is measured by a 

 small quantity <5E which will involve e, f? and <V, where S<a 2 is the increase in the 

 value of <a 2 for the distorted figure. 



The first order displacement of the ellipsoid will be represented by the first terms 



T = c-t'S s , .......... (164) 



of equation (163), and this is supposed to be a possible figure of equilibrium with 

 Sta 2 = 0. The corresponding value of <?E will be of the form 



SE = ae 2 , .......... (165) 



and the coefficient a must therefore vanish if the original ellipsoid was one 

 corresponding to a point of bifurcation. 



If we do not at present assume that a = 0, the value of <5E arising from the second- 

 order displacement (163) will be of the form 



, . . (166) 



in which a single term in / has been taken as being typical of all the terms in the 

 coefficients//. The condition that the displacement (163) together with an increase 

 $<o" in to 3 shall give rise to a figure of equilibrium is that E shall be stationary for all 

 variations of e and / ; it is expressed by the equations 



|(,SE)=;0, |.(<SE) = 0, (/ = /;, &c.) ..... (167) 



Expression (166) is the same inform as that given by DARWIN ('Coll. Works/ 

 vol. 4, p. 349), except that he omits the term ae 2 from c$E ; and equations (167) are 



