122 PROFESSOR KELLAND ON THE THEORY OF WAVES. 



tion from the general equation, by omitting all terms which 'involve the product 

 of two small quantities. 



No such restriction applies to the former of the equations, and consequently 

 we may deduce conditions from that equation which shall give relations between 

 the different values of b. We do not purpose to pursue this investigation. Very 

 little consideration will shew that the first value of b will be much greater than 

 the second, and that, provided we omit powers of f higher than the first, we may 

 neglect 2 even in the more general case. We obtain, by this process, 



(S B + D a n (D + S, ax/I^ +' h $ Vl^tf= 

 (D, + S, a n g) (S + D m aVl^? yhftn (+' h + y h fc) 

 or S n D m VT^?=nD i> S m ^'/ l .... . (15) 



and S, S m (1 - n*W h + D M D OT n A/!^?= D,, D, n A/l^T 2 (^' A) 2 . 



+ S n S mM 2 ^'A + D^S m ^"A , . (16) 



If, for the sake of abbreviation, we write a for 4' h, and ' for 4" A, and 9 for 



- , we shall have 



by means of equation (15) \/t^i?as-Bpf , which, being substituted in (16), re- 



^n Urn 



duces it to 



or S B D M S^I^TO + S 



S,, D,, D\/l^? a + S m D ro Si + S B D n S 

 which is equivalent to 



since SL-D;=S 2 n -D^4. 

 By means of this last equation and (15) we finally obtain 



T)2 -_. a 2 TO PL S H D K q 



The first of these equations gives 



S*, (1 - w 2 ) = n s DJ, + 4 w 9 + (D* m + 4) 



and the second D^= g-^ 



i + M^ 



