LOST POWER IN VERTICAL AND IN COMMON WHEELS. 51 



necessary to overcome this resistance will be (V cos. $ t;) 1 cos. <p, as may be shown as 

 follows : 



Let G B, Plate xxv. Fig. 8., represent the horizontal resistance or force on the paddle ; it is 

 to be ascertained what force in the direction F B will overcome it. Resolve G B into the two 

 forces H G, H B, one at right angles to, and one in the direction of, the radius rod. The effect 

 to turn the line A B about the point A will be the force H B alone ; the force G H, which is in 

 the direction of the line A B, having no power to turn it, its whole action being on the axle of 

 the wheel. It therefore follows, that the force F B at right angles to the radius rod required 

 to retain the point B in equilibrio, or to exert a force in a horizontal direction equal to G B, 

 is G B cos. <p (because the angle G B H = B A I) and consequently equal to (V cos. <}> vY 

 cos. <f>, as already stated. 



Having assumed in this case the same angle of 44 when the paddle begins to act, the 

 mean of the horizontal resistances on the paddle, viz. (V cos. <p v) 1 = G B will be '547, 

 and the mean of the forces necessary to create these resistances or (V cos. $ t-) I cos..ip 

 will be '522 (the force on the lower paddle being 1) ; which, multiplied by 2|, the number of 

 paddles acting, makes the whole power of the engine employed 1'436 times that exerted 

 on the vertical paddle ; or the proportion of the power of the engine employed on the lower 

 paddle '696 ; the mean given by the experiments being '546 : there is therefore a deficiency 

 of -150 of the power of the engine to account for, which I suppose partly due to the greater 

 friction of this wheel, and partly to the paddles not being quite perpendicular in every 

 position in the water, as has been assumed in the preceding calculations. 



COMPARISON OF THE LOST POWER IN THE VERTICAL AND IN THE 



COMMON WHEEL. 



In the action of the common wheel there arises, as we have before described, two kinds of 

 lost power, one from the retrograding of the wheel, and the other from the oblique action of 

 the paddles ; and we are now enabled to estimate the amount of each of them with con- 

 siderable accuracy, and thus draw a comparison of the efficiency of the two constructions of 

 wheel in different states of immersion. 



The expression (V v cos. <J>) 1 represents the whole mean pressure exerted on the paddle ; 

 this, multiplied by the velocity or space passed through by the centre of pressure in a given 

 time, will express the whole power of the engine. In the same way, by the expression 

 (V v cos. <$Y cos. <p, the resolved horizontal resistance, we may obtain the mean effective 

 pressure acting on the vessel ; which, multiplied by its velocity, will express in the same 

 manner the proportion of the whole power which is useful in propelling the vessel. These 

 numbers are computed and arranged within all practical limits in the following table, the 

 original engine power in each case being assumed to be 1. 



In the vertical paddle wheel, as there results no loss of power from oblique action, the ratio 

 of the useful to the whole effect will be that of the velocities of the wheel and vessel : ' this 

 ratio, as will be seen from the experiments, is 3 to 2, or the proportion of effective power is 

 666 of the whole power. 



1 See the note at the end of this paper. 



