424 SEVENTH REPORT—1837. 
is the prolate cycloid, and as the height of the wave increases 
the form approaches that of the common cycloid, becoming 
more and more cusped until at last it becomes exactly that of 
the common cycloid with a cusped summit; and if by any means 
the height be increased beyond this, the curve becomes the cur- 
tate cycloid, the summit assumes a form of unstable equili- 
brium, the summit totters, and falling over on one side forms 
a crested wave or breaking surge. 
6. A wave is possible in forms of channel where the depth is 
not uniform throughout the whole depth. The full considera- 
tion of this subject is reserved for next report. It appears 
however that where the difference between the depth of the 
sides is considerable, one part of the wave will continue during 
the whole period of propagation in the act of breaking, so as to 
show that in these circumstances a continuous wave is impos- 
sible. In other cases the ridge of the wave rises so much higher 
on the shallower part of the fluid as to produce a given velocity 
without exceeding the limits of equilibrium, and in those cases 
the wave becomes possible, and the velocity appears to coincide 
closely with that which we obtain by supposing the wave re- 
solved into vertical elements, eack having the velocity due to 
the depth and then integrating. 
For example, let the form of the channel be 
y =m xn 
x dy = vertical element of area 
as See = the square of the velocity of the element, 
and 
> a? x28 y = the square of the velocity multiplied by wave, 
whence, 
A 2223 = pe oie ge tae 
aula 2 
1 2 
=f —admnartlie 
* 2 
os in mn 
2° n+2 
ant+24 C, 
But since 
mn 
ug Sak Ll 1 
Sfary ras id 
