522 PROFESSOR KELLAND ON THE THEORY OF WAVES. 



that is, if we divide by b cos 6 



-2 6'(e«^ + e-''--)cos2 + 36'(e«^ + e-«^ 

 a 



Now the expression in art. 10. is this, 



-8 6csin^ + <^'(e«-+e— "^) + 26'l-cos2^(e«- + e-«^ 



which is equivalent to 



-2 6^ (e"- + c- "-n cos 2 a = — (e"-' -e" « M 

 an equation identical with the one above. 



21. Let us now derive from the equation the value of c in terms of b^b^ ... . 



1. In the first place, since bo is in every place subtracted from c, never occur- 

 ring in any other way, we derive the following important conclusion : 



That a progressive motion of the fluid does not affect the velocity of trans- 

 mission of the undulation relative to the position of corresponding particles ; in 

 other words, the velocity of transfer of the undulation is exactly equal to the sum 

 of the velocity of progression, and that of undulation in a fluid at rest. 



2. If we make 6=0, we obtain 



(c - bj Irbr (e^ "■ '' + e" '" « *) 



which equation gives c-b„. 



The two values so obtained will be equal, but will have opposite signs, since 

 the term which involves the first power of c— 6„ does not contain any cosines. 



This cu'cumstance that the sines are combined with the odd powers, and co- 

 sines with even powers of c-b„ is very remarkable, and it is probably connected 

 with the relation existing between the quantities b,,b.2 .. . , but we shall not at 

 present enter into a discussion of the subject. 



If it be thought more simple to obtain (c—boj in terms of o,, a^ . . . , than in 

 terms of b,, b., ... , this can be at , once effected by means of the equations in 

 art. 14 ; the result being ; 



