320 MR. P. W. BARLOW ON THE LAWS WHICH GOVERN 



In order to get an expression for the resistance in a horizontal direction, or that 

 part of the power which is effective in propelling the vessel, C G must be resolved into 

 the two resistances G I, C I, of which the former is (V — 2; cos <p)2 cos (p ; and it is to 

 be shown that a mean resistance which would act uniformly through the arc «p, so as 

 to be equal to this variable action, will exceed that of the mean action of the lower 

 paddle ; while in the new wheel, the mean resistance is less than that of the lower 

 paddle, and hence the great difference in the mean numbers in the Table. 



In the new wheel it has been already stated, that the paddle enters the water 

 nearly in a vertical position ; and in order to simplify the investigation, I consider it 

 to be truly vertical in eveiy position, which is so near the truth, in that part of the 

 revolution where the action of the paddle takes place, that the results will be but 

 slightly affected. Let C D, figure 4, be any position of a vertical paddle moving at a 

 velocity V, in the direction F B of a tangent to the circumference. Then by resolving 

 this velocity into two, one at right angles to, and one in the direction of, the paddle, 

 we find the velocity with which it meets the water at right angles to its face, to be 

 V cos <p, <p being as before the angle of inclination of the radius A B with a vertical. 



The resistance opposed to the vertical paddle when the ship is in motion with a 

 velocity V, will therefore be (V cos <p — v)"^, so that in the vertical paddle, when V cos (p 

 is equal to v, no resistance is opposed to the engine, and when it is less the paddle 

 opposes a resistance in a contrary direction ; and it is sufficiently obvious that the 

 resistance in every position in this case is less than when in its lowest position, while 

 in the old wheel it is everywhere greater, at least within practical limits, which fully 

 accounts for the difference in question. 



It is observed above, that the horizontal resistance of the oblique paddle is always 

 greater than in its vertical position within the limits prescribed by practice. Let us 

 examine what the actual limits are, by finding, when with given velocities V and v, 

 (V — V cos (py cos ^ is a maximum, or when 



y^ d cos (p — 4V V cos ^ . d cos <p + 3 \^ cos v 2 1^ cos ^ = 0. 

 Whence 



„ 4Vcosa V^ 



cos <p2 _ Z- ~ _ 



and 



cos?. = ^. 



It depends, therefore, on the relative velocities of the wheel and vessel. 

 When V = 5, i; = 4, then <p = 65° 33' 



V = 4, «; = 3, (p = 63 37 



V = 3, z; = 2, (p = 60 0. 



These results at once account for the ratio of the power of the engine to that of the 

 resistance on the vertical paddle being greater in the old than in the new wheel. For 

 it appears, contrary to the usual opinion, that not only the total resistance to the 



