322 THE MECHANICS OF THE EARTH's ATMOSPHEKE. 



in which, 



F =( \^ ^ '*:^ \I^ n3\ 



and putting n' -\-'2 for ;/' in (12) we get 



A, 





_^.+-3 





This becomes by substituting for _ iL+r its value derived from thepre- 



A„,j.n 



ceding expression 



K — " -I F F 



In like manlier we get generally 



iC^,=^^,,+,F,,^,F„^, .... F„.^, . . . (14). 



in which the values of the factors i^„ +2, -F„ +4, i^„ +,, fire given by (13) 

 by adding 2, 4, and l respectively to n' in that expression. 



Now, all these factors are finite, and hence putting now K„ for its 

 equivalent, ii„ +, and A„ for A„,4-,, we have 



^n'+-i J^n = a finite quantity 

 since by (10) 



^^^+2 A,^ = a finite quantity. 



From (13) and (14) it is seen that any coefficient, taken without 

 regard to signs, 



^.+. <z'r^"'+^ (^^^ 



since when this condition is satified all the factors i^,,^ o, -F,,.,. 4, . 

 • . F„, + , are less than unity. Therefore, we have, putting ?t for w' j- /, 



:^-^, Ik . . . < ^T+3 f'-^n, or by (!()),<- Jii^(l + ^,"' A,. ) (17) 



when /S'>G (IS) 



since this is what (10) becomes when i is infinitely great. This is simply 

 the limiting condition in all cases, and the first number of (17) is gen- 

 erally less than the second when /i is considerably less than G. 

 We have from (3) 



« = >T A', = P„ + (,>,, . (19) 



