522 PROFESSOR KELLAND ON THE THEORY OF WAVES. 



that is, if we divide by b cos 6 



-2 b' [f - + e- « -) cos 2 + 3 If {e" - + e- « -") 

 a 



Now the expression in art. 10. is this, 



-8 6 csin0 + «'(«"*+«""") + 2 4' 1-008 2 6(6"- + «-«') 



— b'(e + e —e + e ) = — (e —e ); 



^ a 



which is equivalent to 



c'ie'"' +e-''')-8besm6-b'(e^" +t-''"') +36'(e"- +e-"-) 

 -26'(e''' +e-"')cos26 = — (e"~-e-"*') 



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 h„ is in every place subtracted from c, never occiu-- 

 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 progi-ession, and that of undulation in a fluid at rest. 



2. If we make 0=0, we obtain 



{c - bj 2 r br (e'- « '' + g- <■ " *) 



+ 2 r J, 6, 6, { - («•'"" * + e-f" '' ) - (/■'■- - *■) "• '' + e" 



-/— 2r«A 



a 



which equation gives c-6„ . 



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

 the term which involves the first power of c—b„ 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-6„ is very remarkable, and it is probably connected 

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

 present enter into a discussion of the subject. 



If it be thought more simple to obtain (c-6„)- 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 : 



