318 Prof. Darwin, On the perturbation of a comet [Mar. 7, 



4 



We have ^- 4 = 1 + 4 -^ cos 0, 



and therefore ^ = (-^-J ? (l + 4 £ cos *) 



F 2 _{ M V^Vi , a P 

 P 

 Equating F*/C* to JP/®*, we get 

 'SVfS + m 



1 + 3 cos 2 5 



\p) ~[m) \M+mj 



- 4 cos (1 + 6 cos 2 6>) -£J 



: 8)*££>*ci + w«* 



_2 cos (1 + 6 cos 2 0) p 

 5 " 1 + 3 cos' R 



Thus the equation to the surface is approximately 

 R_ /S\ifS+m 

 P 



-Of tiffin-"* 



2(M\i {3f+m\i cos (1 + 6 cos 2 0)\ 



5\8J \S + mJ (1 + 3 cos 2 0)™ J ' 



It is usually the case that m is negligible compared with M, 

 and that M is also small compared with S, and in this case we 

 may write the equation with sufficient accuracy 



!-(J)*<i+w«*. 



Laplace gives a formula for the radius of the sphere of activity 

 which is virtually derivable from the above investigation on the 

 special hypothesis that the three bodies lie in a straight line. 

 Thus he puts equal to zero or 180° and finds, 



But to find the true mean value of (1 + 3 cos 2 0) TT , we must 

 estimate it all over the sphere. 



Now 



~ |T(1 + 3 cos 2 ey& sin 0d0dj> = T (1 + 3^ 2 )" dx. 



This integral evaluated by quadratures, is found to be equal 

 to 1-063. 



