Mr T. Tredgold on Steam-Boats . 249 
not to be increased more than can be avoided, because there is 
not then time for the water to flow between them, so as to afford 
a proper quantity of reaction, neither do they clear themselves 
so well in quitting the water. If we suppose W L, Fig. 2., 
Plate VII. to be the line the water would assume, when at rest, 
the most favourable arrangement, with the smallest number of 
paddles, appears to be to make the paddle A of the wheel A 
just entering, when the preceding one B is in a vertical posi- 
tion, and the one C quitting the water. This arrangement al- 
lows time for the water to flow between, and for it to escape 
from the retiring paddles. If a smaller number be employed, 
there will be a short interval, during which none of the paddles 
will be in full action. The utmost variation will be between the 
positions of the wheels A and B, Fig. 2., and an intermediate 
position is shewn by the wheel C. I have not attempted to 
represent the actual state of the surface of the water during the 
motion of the paddles, for, unless it were done with accuracy 
according to nature, it is better undone ; but the form of the 
surface will not materially affect the conclusions. 
To determine the radius of the wheel, or the depth of the 
paddles, when the number of the paddles is given, becomes an 
easy problem, when the preceding conditions are to be adhered 
to. For, put AO (Fig. 3.), the radius = r, and x — the depth 
360° 
A a of the paddles ; and n their number. Then 
n 
the 
angle AOB contained between two 
paddles, and r cos. =£» 
O a ; the cosine of the angle, being the depth from the centre 
of the wheel to the surface of the water ; and, 
360 
r cos = r — ar, or 
n 
r ^1 — cos = x = A the depth of the paddles. 
Also 
x 
cos 
360 
— r — Ao, the radius of the wheel. 
n 
From these equations we have the following rules, viz. To find 
the radius of the wheel, when the number and depth of the pad- 
dles are given. Divide 360 by the number of paddles, which 
