508 PROFESSOR KELLAND ON THE THEORY OF WAVES 



xdtr \dt) ^^i^g tli6 co^ipl^te differentials of ?< and ?7 divided by fl^^. 

 These equations, again, give as their result, 



d /du\ _ d (dv\ 

 'dy \di) ~ Jz \df ) 



which being combined with the equation T" + ^~^' ^^^ motion will be ob- 

 tained. 



-, /du\ du du du 



(dv\ dv dv dv 

 dt) dt dx dy 



But if the wave be oscillatory, we may assume for u and v a series of terms of 

 the following form : 



'»'=f{y) • sin -^ {ct-x) 

 v = h y . cos -T— (c t—x). 



For it is obvious, without any calculation, that, since ^^ + ^^=0, if 3- involve 



*' dx dy dx 



27r 

 cos -y^{ct—x), II must do so too, and consequently v will contain only cosines of 



quantities, of which u contains sines. By substitution in the equation j~ + j~=^- 

 we get the following result : 



27r 27r , . ^ 27r , , „ 



or F> = ^/>, (1). 



2'7r 



Let -^ be denoted by « 



2'7r 



I 



then the values of \Tn ^^^ ( ^7) become (retaining only this term of the value 

 of m), (^j = <^fy cos Q {c-fy sin &) +fy Fy sin cos 6, 



i — \ = —ttFy sin 6 (c—/y sin 6) + F'y Fy cos 2 ^ . 



See the references in Art. 6. 



