340 Prof. G. H. Darwin. On the [Nov. 25, 



Thtis tile constants G and B 3l 1? 5 , &c, are all expressible in terms 

 of B v 



We may remark that if 



-i-h pB^ifti!, or B_ Y = -2E, 



then the general equation of condition in (7) may be held to apply for 

 all values of i from 1 to infinity. 

 Let us now write it in the form — 



f^ = l-7^/ 3 + ^|^ (8) 



-°2i-l 2 -°2i-l 



When i is large, B 2iJrl jB 2i _ l either tends to become infinitely small, 

 or it does not do so. 



Let us suppose that it does not tend to become infinitely small. 

 Then it is obvious that the successive B's tend to become equal to one 

 another, and so also do the coefficients -5 2 i-i~/ 2 -^2i+i i n the expression 

 for dujdfi. 



du ^ . M 



^ u jy[ 



Hence — =—L Vl — +— 7=, and therefore x is infinite when 



dO r> ^ 



fx = 1 at the pole, and d^jdt is infinite there also. 



Hence the hypothesis, that B. 2i+1 IB 2i _ l does not tend to become 

 infinitely small, gives us infinite velocity at the pole. But with a 

 globe covered with water this is impossible, the hypothesis is negatived, 

 and B ii+l /B 2i _ 1 tends to become infinitely small. 



This being established let us write (8) in the form — 



B<li-\ •" 2i(2t+l)^ 



Hence __=£-)-- where L, M are finite, for all values of u. 



1 — ft 4 



2t(2t+l) 



(9) 



By repeated applications of (9) we have in the form of a continued 

 fraction 



7? — 1 b \+ - & Pf 7 - # iT&c. 



-*>2.-l 2»(2H-1)^ [ (2HjHg!±8r_ I (2i+*)(2i+5)^ | 1 



^2i-3 x 2 i(2*+l)« / ' (2t+2)(2i+3/ ^ (2i+4) (2i+5)J t* 



... (10) 



And we know that this is a continuous approximation, which must 

 hold in order to satisfy the condition that the water covers the whole 

 globe. 



Let us denote this continued fraction by — JV*. 

 then, if we remember that B^— — 22£, we have 



