272 



Mr. G. H. Darwin. 



[Mar. 18, 



evolution, whilst the obliquity should not increase to any very large 

 extent. But it must be remembered that we are here only treating a 

 planet of small viscosity, and it will appear, in the unpublished paper 

 above referred to, that the rate of increase or diminution of the eccen- 

 tricity is very much less rapid (per unit increase of x) if the viscosity 

 be not small, whilst the rate of increase or diminution of obliquity 

 (per unit increase of x) is slightly increased with increase of viscosity. 

 Thus the observed eccentricities of the orbits of satellites and of 

 obliquities of their planets cannot be said to agree in amount with the 

 theory that the planets were primitively fluids of small viscosity, 

 though I believe they do agree with the theory that the planets were 

 fluids or quasi-solids of large viscosity. 



We now come to the second case, where h is less than 4/3*. The 

 biquadratic having no real roots, we may put 



^-/^ 3 + l = [(ro-a) 2 -fy3 2 ][(«- 7 ) 2 -f £ 2 ]. 

 It has already been shown that a is negative, and 7 greater than jh. 

 Let a=§h—a v <y=<Yi+}^ 



Then by inspection of the integral in the first case we see that 



•_ , CO" 7)- + 3 2 ]8(v 1 2 +? 2 ) 



J~ A ha, X 



[(^- a ) 2 +y3 2 ]8< a > 2 +e 2 ) 



[h/3 , x—a . hd & — 7I 

 — arc tan -f arc tan . 

 4(^ + /3 2 ) /3 U( 7l 2 + 6 2 ) r J 



The rest of equations (21), which express the other elements in 

 terms of j and x, remain the same as before. 

 By comparison with the first case, we see that 



& 1 -oc 1 (x-oc)+^ + 1 7l (aj_ 7 ) + g2 



+ 1 2(V + /3 2 ) 0*-*) 2 +/3 2 2( 7l 2 + S 2 ) (*- 7 ) 2 -R 2 



On multiplying both sides of this identity by x 4 —hx 3 -{-l. and equating 

 the coefficients of x s , we find 



2( ai 2 + /3 3 ) 2( 7l 2 + a 2 ) 



Therefore - — = ^ 



Thus when x is equal to +oc 



