PROFESSOR KELLAND ON THE THEORY OF WAVES. 529 



/ ds du\ „ / 1 2\ ^*^ . ^* ^ 

 Y dx dx) V 2 / dx^ dx'' ^ 



dx dx) 2 dx ' 



and mass moved = K a ^ 



= ap Fi/ = aQFs 



n <^* dul„ds(bs,^, 



therefore movmg force =-^rf^ + «^+2''-^F7 ^^^ 



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

 great facility. 



Also, as in other cases, 



'dx + d a,\^ 



^doc + day _ / du \^ 

 V 1} ) ~ \ 'dx"') 



and 



da _du 

 dt dx 



du 1 da. 1 dUL 



dx a, dt K d t 



Now K = F. 



dJL -r^, ds 

 " dt dt 



, ds 



— (bs.-r- 



^ dt 



dw _ (p s ds ,„, 

 •'• d~x Fl'Tt ^^ 



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 



„ 2/— 3 J d)s S 1 



and Ft/=-Vm.y^ and ^^=2 ^ ' 



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



du _ ds du 1 2^*3 1 

 dt dx dx 2 dx 2 s 



or -^r-bc .cos 6——— b^ sin 6 cos = 



A. A 



P, „ n b^ sin^ 6 -^— a cos 6 



2^- ^ 27r., . /J /J 3 A 



-r-affC0SC7 ^- sin a COS t7 — -. ; : — a 



A"^ A 4 h + asmo 



therefore ag—bc=0 by equating the large terms as in other cases. 



VOL. XIV. PART II. 4 X 



