252 
Mr T. Tredgold on Steam-Boats. 
Then the resistance to be overcome to give the boat the velo- 
city v 9 is, when the motion is with the stream, u 2 : ( v — c) 2 : : 
a : 
a (v — cf 
u 2 
And, when the boat moves against the stream, as 
: a 
u* 
Hence, the power in either case is expressed by 
gp( + c ) 2 
U 2 
The upper sign to be attended to when the motion is with 
the current, and the lower sign when it is against it When 
c, the velocity of the current, is nothing, the result is the same 
as before. But the resistance in still water is not the mean 
between the resistances in the direction of the current, and 
against the current ; consequently, the mean rate of a boat, 
which alternately goes with and against a current, must be 
less than the mean rate in still water. The mean resistance 
Cl *v (eft — J- c 2 '\ 
is -, while the resistance in still water is only 
u j 
CL ^ ^ 2 
~ 2~5 and the difference between these is 2 - ; a quantity de- 
pending on the velocity of the current, and, for any particular 
case, should be calculated from the mean motion of the current. 
When a boat advances with a current, the velocity with which 
the paddles act on the water will be 
V + c — v ; and when the boat moves against the current, it 
will be V — c — v ; consequently, in either direction it is 
V -f- c — v ; and the force of re-action (V -f c — v ) 2 . But the 
effective resistance of the boat is as V + c — v : v : : (V + c v) 2 
• ^ (V ”h c 5 &nd its effect m a given time is a maximum 
when v 2 (V + c — v) is as a maximum, that is, when 
v + 2c 
2 
, or when V = 1 -5 v + c. Also, 
(V + c) 
~~1r~ =v * 
2 V 
When c = o, or the boat moves in still water — v 9 the 
o 
same as before, and the mean between moving against and with 
2V 
8 
the current is = v also ; therefore, where the velocity cannot 
