PROFESSOR KELLAND ON THE THEORY OF WAVES. 529 



(ds ■ du\ -r, I \ ,\ d s , d s . 



^d-x"'d~x)''^'-V'-2''TTxf"-"''d-x''f"' 



(dsdu\„l„ds, 

 ff j-~u-j-)aFs + ^au- -—(ps 

 dx dx) 2 dx 



and mass moTed = K « p 



= a.q Fi/=aqFs 

 n <^* du\,ds(bs,-. 



therefore movmg force =-^^^ + "^^+2" • rf^ • F7 (1) 



This is an expression which can be applied to different shaped channels with 

 great facility. 



Also, as in other cases, 



and 



Now 



By means of these equations the velocity and motion are discovered. 



27. Ex. Let us take one example for the sake of illustration. 

 Suppose the canal to have the parabolic form, then 



and ^y=-^m.y^ and ^^=2 » ' 



By the substitution of this value in equation (1), we obtain 



du _ ds du 1 ,dsS 1 

 dt dx dx 



27r^ a 2ir, 



dt' ^dx^^dx'^2^ dx2s 



or -^— 6c. cos 5 — :^— b'' ein 6 cos 6 = 



A. \ 



o _ r. o 6" sin^ 6 -r— a cos 6 

 -r- a a COB a — ^^ 6' sm d cos d — 3 r -. — 3 



therefore ag—bc=(i by equating the large terms as in other cases. 



VOL. XIV. PART II. 4 X 



