ON THE GENERAL THEORY OF THE STEAM ENGINE. 191 



II. Power ex- ("by the Engine p V cos. </> cos. (<f> e) . dt ~] 



pended hori- J on the Fluid = p (V cos. <j> v) cos. (</> e) . dt > . . . . (d). 



zontally [_on the Vessel =p v cos. (<f> e) . dt 



III. Power ex- fby the Engine i 



i-u -c-i j f = P V sin. <f> sin. (<i e) . o# 

 pended verti- < on the Fluid J V . . . . (e). 



cally Ion the Vessel = o 



In each case it will be observed that the power expended by the engine is made up of the 

 powers expended on the fluid and on the vessel ; it will also be observed that the entire 

 expenditure is made up of the horizontal and vertical expenditures. 



The whole power expended by the engine being p V cos. e . dt, and the useful power, or 

 that expended horizontally on the vessel, being p v cos. (0 e) . dt, the proportion of the 



power that is effective is ^- -, which evidently increases with the angle e. It attains 



V cos. e 



the value of unity when e attains the value according to the equation (b), or when the paddle 

 moves through the water edgewise, and produces no effect. If e be taken greater than the 

 value so determined, the effective velocity will become negative, and the paddle will act in the 

 contrary direction and retard the vessel. It is wrong, however, to suppose, as is sometimes 

 done, that the misdirected action of the vertical paddle in such a case constitutes a dead loss 

 of power, since it is plain that when the paddle ceases to propel the vessel, it will at the same 

 instant cease to put the engine to any expense of power ; and when it impedes the vessel, it 

 will, at the same time, assist the efforts of the engine, and thereby increase the action of the 

 other paddles. 



By integrating the preceding expressions (c) (d) (e), we shall obtain the respective amounts 

 of power developed by one paddle during any finite time, or throughout any proposed finite 



a 



distance. For conciseness let N dt denote any one of them. Then / N dt will give the 



o 

 power expended during each half revolution of the wheel, supposing the motion of the paddle 



to begin or end at the lowest point D ; or since dt = ^ , 



a. 

 Power expended by one paddle in each "i k /* -^ , , 



half revolution / V J 







a. 



Power expended by all the paddles in "l _ 2 m k f* ^ , , 

 each half revolution / V J 







Now, the circumferential distance traversed by the centre of pressure, in half a revolution, 

 being IT k, and the velocity per minute being 60 V, the number of minutes elapsed will be 



-. Therefore, dividing the last expression by this number, we have, 

 60 V 



a 

 Power expended per minute by "1 120m /* ,. , ., 



both wheels J TT J 



