ON THE CENTRE OF PRESSURE OF A PADDLE. 49 



The actual amount of effective pressure at any angle may be ascertained in the following 

 manner. Let A B, Plate xx v. Fig. 7., be any position of the paddle board of a vessel in motion ; 

 V being the velocity of the wheel, and v that of the vessel, and <f> the angle of inclination of 

 the paddle board with a vertical line. Let C D represent the velocity V at right angles to the 

 paddle, and E C that of the vessel in a horizontal direction. Then it is evident, that C F, 

 which is the resultant of these velocities, will represent the velocity and direction of motion of 

 the paddle with respect to still water. Resolve F C into the two velocities F G, C G, one at 

 right angles to, and the other in the direction of, the paddle ; of which the latter is lost, while 

 the former will represent the velocity with which the paddle meets the water in a direction at 

 right angles to its face : then CG or HF = EF EH = V v cos. <p ; consequently, 

 (V v cos. <p) J will represent the whole resistance which the paddle opposes to the engine at 

 any angle <f. 



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

 is effective in propelling the vessel, C G must be resolved into the two resistances C T, GT, 

 one in a horizontal, and the other in a vertical direction. GT or the effective propelling 

 force will therefore be (V v cos. p) 1 cos. <p : and it will be found in all cases which occur 

 in practice, that the velocities V and v are such that this expression is greater than (V 1>)% 

 or that of the resistance on the vertical paddle. 



In the case of the new or vertical wheel at any angle <f of the paddle rod, the resistance 

 will be (V cos. <p r) 2 ; so that in this wheel, when the angle is such that V cos. if is equal 

 to v, no resistance is opposed to the engine ; and when it is less, (a case I have witnessed in 

 practice,) it opposes a resistance in a contrary direction, or is a direct impediment to the 

 motion of the boat when first entering the water. It is, however, sufficiently obvious, that 

 the resistance opposed to the engine in this case is less than when in its lowest position, while 

 in the old wheel it is every where greater : it will also be readily seen, with such velocities 

 V and v, as usually occur in practice, that not only is the resistance to the engine greater 

 in the oblique paddle, but that more effect is produced by them in propelling the vessel, 

 unless the immersion of the wheel is very great, as is sometimes the case in stormy weather, 

 when of course the angle may be such that the resistance is almost entirely in a vertical 

 direction. The angle when the effective resistance is the greatest may be found as follows : 



(V v cos. ip) 1 cos. <p is the maximum when 



V 1 d cos. $ 4 V D cos. if d cos. <p + 3 v 1 cos. Q 1 d cos. $ = ; 



4 Vcos.<t V 2 



whence cos. <* 5 = .-, j ; 



3 v dv 1 - 



V 



and cos. <J> = .-. ; 

 3 v ' 



hence, with the velocities such as form the limit of practice, viz. 



V = 5, v = 4, <p, or angle of greatest effect, is . . 65 33' 



4 3 63 37 



3 2 60 



It should be stated, however, that although an increased propelling power is obtained at 

 the above angles, it is not to be understood that so great an angle is practically advan- 



S 



