270 Prof. Challis on the Mathematical Theory 



The radius of the lower surface of the ocean being b> the condi- 

 tion u = for this surface gives 



2^-3c 3 6~ 4 = 0, or -^ = -£- 



From the above values of ~ and ~ it may be shown, just as 



ma 

 in the previous investigation, that ty (t) = vr + Ga + ^3> and is 



therefore a constant. 



Again, by substitution in the equation obtained by differen- 

 tiating (y) completely with' respect to t, and equating the left- 

 hand side to zero after putting a for r in the other side, there 

 results 



o=- 2 G Cl ( a -5) + ^ + v Cl (^^S). 



Hence 



Sma/jb 2/^2 

 4&R? + ~Ga~ 



a b G 



o+© 



Let 6 be the value of 5 which causes the denominator of this 



ll a 

 fraction to vanish. Then, putting co for ^-, it will be found 



that this denominator is equal to 



fl-*o(i-g- 



We may now employ the arbitrary quantity c_ 2 to get rid of the 



factor 1 7-, by which the result would otherwise be vitiated. 





 It is clear that, since for this purpose c 2 must be negative, we 



must assume that 



Smafj, 2/u rc^ 



Sma/jb /, b \ 



4GR 3 ' Ga 



for if we put C ( 1— -7-) in the place of the simple factor 1 — j-, 



C being any arbitrary positive quantity, the two sides of the 

 above equality could not be identical. Hence it follows that 



Sma/jb 

 4GR3 



r b W- b 3 b 4 \* 



