530 PROFESSOR KKLLAND ON THE THEORY OF WAVES. 



And from equation (2) 



d u __ 3 ds 

 dx '2s d t 



(A + osin6)6cos6 = ^acos6(c — 6sin6+ ... 



h, 3 



n b = -=a c . 



If we put for h this value in the former equation, we get 



3 ac' 



or the square of the velocity in a parabolic vessel is 5 of its value in a rectangular 



o 



one. 



28. We may readily deduce the more general results, viz. 



aff-bc = (1) 

 6 cos 6= a cos ^(c — 6sin 6)^- 



.= «c|i (2) 



a^=ac^-^ 



'-=^<pk 



<p/i 



w 



h 



area of vertical section 



breadth at surface 



29. The investigation supposes the curve a continuous curve, but, except in . 

 extreme cases, it will apply equally well to others. For instance, it wiU apply 

 to the cases examined by Mr Russell, viz. when the areas are trapeziums. We 

 will apply the formula to one or two of these experiments of Mr Russell, and 

 then quit the subject. 



In the channel M, the breadth is 12 inches, and the depth of the triangular 

 part 4 inches : therefore area of triangular section equals 24 inches. And if we 

 take the height to the centre of the wave as the height corresponding to h, we get 

 for experiments xc, xciii, mentioned p. 444 ; 



A = 6.21, 

 area of parallelogram =26.52, 



therefore area of section =50.52 



1 



