508 PROFESSOR KELLAND ON THE THEORY OF WAVES 



dp 



d. ?(-(47)) 



dp 



(clii\ tdv 



^=?{-^-(£)} 



(rf?) ' \dl) ^^^^S *^® complete diflferentials of *« and v divided by dt. 

 These equations, again, give as their result, 



d ldu\ _ d (dv\ 

 dy \dt) ~ dx\dt) 



du dv 



which being combined with the equation w~'^rf~~^' *^® motion will be ob- 

 tained. 



du du du 



dt dx dy 



_T (du\ U H. I, M. It M 



(dv\ dv dv dv 

 -l-\=-;- + u-- + v-- . 

 dt/ dt dx dy 



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

 the following form : 



"=/(y) • sin — {ct-x) 

 » = F^.cos-^— (et—x). 



A. 



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



'' dx dy dx 



2'7r 

 cos -^ (c'-a-), V 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 2^ , , . 27r , „ ^ 



— -^ *'*'*"ir K'^^~*)ty + cos -—{ct—x) F'y=:0 



or F'.y-^/y, (!)• 



217 



Let ^ be denoted by « \ 



'^JL(ct-x)hye i 

 then the values of f ^ j and \jj) become (retaining only this term of the value 



of M), \-^\ - o-fy COS 6 {c-fy sin &) +fy Fy sin cos 6, 



{il\ = -aYy sin 6 {c-fy sin 0) + Y'y Fy cos ^ . 



* See the references in Art. 6. 



