AND THE NEBULAR HYPOTHESIS. ig 



Denoting as before the density of the crust at its inner surface (r = a) by </, aud 

 putting <T O , the density at the outer surface (r = c) equal to zero, we have 



P -'/ T / > \ i W 1 

 \j a Ji /f ( K a, a) = <r , 



whence, on elimination of C , 



Equation (72) still gives 



-''. J, 



_ j _ c sin 



a sin (*' 

 J..V, (*V, -ft) = 0, 



(90) 



as the condition for points of bifurcation, and when n = 2 (the only case which 

 appears to be worth examining), this reduces to 



tan (<c'e+/8) = K'C, 



(91) 



in which /3 is given from the table on p. 479. The procedure is to find /< from equation 

 (91), and hence calculate the value of utfeiro-' from equation (90). The results for a 

 few values of K 'a are given in the table following (the last column is explained 

 later) : 



The obvious remark must at once be made that prol>ably all the values for 

 are too large for results obtained by the neglect of w* to be accurate. But apart from 

 absolute accuracy there is an obvious tendency for the value of K^/STO-' to fall off as *'u 

 increases for lower values of c/a the symmetrical configuration becomes unstable ( >r 

 lower and lower values of w"/2w'. For /a = 100, the value of w*/2T<r' is 0'0104. 



24. Against this, it must be noticed that the value of w'/'2ir<r f is of very slight 

 importance ; what we are concerned with is the ratio of w'/'Z-ir to the mean density of 

 the whole mass. For a very rough calculation, we may assume the mean density of 



VOL. ccxni. A. 3 Q 



