190 Mr. G. H. Darwin on a suggested Explanation 

 Hence (1), (2), and (3) lead to the following equations: — 



l{ i+ &<*-»} **"*> • • • (4) 



p« 3 =— ? ... (5) 



j^-i , J l)Ma 2 = C-A. . (6) 



4tw I .(gay _ u J 



3k(qa — 1) 

 If during contraction qa remains constant, and if the coefficient 



n 



of Ma 2 n in (4) be called y, and that of -j in (6) be called 



/3, then it will be found that 



n^ 1 ;_ 7 127TfjL P 2 dt 



Hence, remembering that GVj = H, 



-7- log tan Q— -~- »* 



f//0 to r27T(JLyp' 



Integrate, and let D, I be the present values of p and ; then 



, tan 6 _ p/3 /.. D\ 

 g tanI - 127r/J>yV p/' 



If we assume that ga has always the same value as it now has 

 in the case of the earth*, 



7 = -3344, £=-9507, and ^ =2-8433. 



If during contraction the planet were always homogeneous, 



8 15 



the factor — would be replaced by — , or 3*75. 



Let K stand for 2*8433, or 3*75, as the cnse may be; let 

 Q = ff2 + ^ m/2/1 , \ 5 ^ P De the periodic time of a pen- 



* In determining the precessional constants of Jupiter and Saturn, La- 

 place assumed that their law of internal density was the same as that of 

 the earth. The assumption is, I believe, unjustifiable; but it will give 

 sufficiently good results for the present purpose. The limiting value of 



— , when the surface-density is infinitely small, and if the Laplacian law 



stiirholds good, is 1"99. See ' Monthly Notices of the Eoyal Astrono- 

 mical Society,' December 1876. 



