ON THE GENERAL THEORY OF THE STEAM ENGINE. 189 



slow. This disturbance has already been partially alluded to in the foot-note at page 311 of 

 this work, and it will no doubt, to a certain extent, vitiate any calculation of the power of an 

 engine which does not include its influence, however exact it may be in other respects. Be- 

 sides, independently of this disturbance, the assumed law that the resistance varies as the 

 square of the velocity, cannot be regarded as satisfactorily established, since another agent, 

 of considerable influence, is to be found in the waves produced by the action of the vessel on 

 the fluid. For these reasons we are inclined to doubt whether, to practical men in particular, 

 the general accuracy of the calculations can reach that degree of minuteness which will com- 

 pensate the labour incurred by the complexity of Mr. Mornay's analytical expressions. His 

 investigations cannot fail, however, to be highly esteemed by such of our scientific readers 

 as may be desirous of thoroughly examining the theory of this interesting and important 

 subject. 



We here propose, in the first place, to explain precisely how the power of the engine is 

 distributed in the general production of effect ; then to show how the determination of the 

 power of a wheel, by means of the centre of pressure, may be materially simplified for prac- 

 tical calculation ; after which we shall add a few remarks on the resistance of fluids. 



Suppose a force Q to intervene between any two objects A, B, tending to their separation, 

 and to act simultaneously upon both of them in the opposite directions, according to the 

 property of action and reaction. Then, if da, db, denote the indefinitely small distances 

 described in an element of time, and c the mutual distance of A and B, by the principle of 

 living forces the quantity of power expended on A will be GL da, that expended on B will be 

 Q, db, and the entire expenditure of power will be Q da + Q, db = Q dc. In this manner we 

 shall find no difficulty in resolving the action of a paddle wheel into its distinct effects on the 

 vessel and the fluid. 



In the annexed diagram let C be the centre of the wheel ; A D B 

 the arc which passes through the centres of pressure of the paddles, 

 and m n any position of one of them when immersed in the fluid. 



Let V denote the circumferential velocity of the point A round the centre C, in feet per 



second. 



v the velocity of the vessel in feet per second. 



k C A, the radius of the wheel to the centre of pressure, and estimated in feet, 

 a the height of C above the surface of the fluid, and estimated in feet. 

 <f> the angle A C D of the radius, with the vertical. 

 the value of <f> when the point A enters or quits the fluid, 

 e the angle re A C of the paddle with the radius. 

 m the number of paddles on each wheel. 

 s the surface of each paddle, in square feet. 

 S the surface of the m paddles on each wheel, = m s.' 

 p the pressure on the paddle, in fes., supposed to be concentrated at A. 



Then, the angle of the float with the vertical = < e ; the velocity V resolved in the direc- 

 tion perpendicular to the surface of the paddle = V cos. e ; that of v = v cos. (<f> e) in the 

 opposite direction ; and therefore the effective velocity of the paddle through the water per- 



2 b 



