202 



Professor Baker, On the general theory of the 



from the expression of the increment of the function H, for varia- 

 tion to the second order, in the case of the ellipsoids, it is known 

 that Cg is negative, and 6V^> is positive; (these are in fact the 

 "coefficients of stability" corresponding to Lame functions of one 

 root, in the former case lying between the negative squares of the 

 greatest and mean axes of the ellipsoid, in the latter case between 

 the negative squares of the mean and least axis). Thus (3), we 

 infer, by algebraic methods only, that a sufficient condition for the 

 instability of the pear-shaped figure, if we assume the preceding 

 ellipsoids stable, is the single condition 





or ^n -f- 2 



{n)\2 





>o. 



m 



,2,i 

 n =0,2,s 



Darwin {Works, iii, 378) computes 



N = - -000235513. 



It is easy to show, from what precedes, that, if Sco'^ denote the 

 increment of co^ in passing from the ellipsoid to the pear, we have 



Nx^ + 





or 



iVa;2 + I8a;2 b + c 



9, 



I 



m 



c, n 



= 0, 



)- 



0. 



Thus a sufficient condition for instability is Sco^ > 0. 



This again, we see algebraically, involves that the increment SI 

 of the moment of inertia, is negative; for, if we put D for the 

 coefficient of ^Sco^ in the last written equation, we easily find 



81 = 



D 





As §a>2, 81 are then of opposite sign it is evidently desirable, if 

 possible, to calculate N independently and not from these, as do 

 Darwin {loc. cit., p. 379) and Jeans ( Cosmogony, 1919, p. 101). 



Mr Jeans {Phil. Trans. A, ccxv, 1915, 76, 77; Cosmogony, 1919, 

 p. 92, § 94) appears, if I understand him aright, to hold the view 

 that an expression such as that above taken for 8H, does not suffice 

 to enable us to draw inferences in regard to the stability, and in 

 particular that there should be present therein a term in x^. And 

 he seeks to find in this way the explanation of the difference in 

 conclusion of Sir George Darwin and himself. I cannot agree with 

 this view; if the expression for 8H is accurate as far as it goes, it 

 appears to suffice for forming a judgment, though hypotheses as 

 to the relative order of smallness oi x, y, ..., should clearly not 

 form part of the process for calculating 8H. 



