ON PADDLE WHEELS. 121 



is, therefore, in any given position of the radius moving in the direction of the radius itself, 

 which consequently lias no angular velocity at this point. The whole jadius may thus be 

 conceived to move in its own direction, and at the same time to revolve about the point F. 

 If we call p the radius of the rolling circle, and < the angle A C D contained between the 

 given radius and the vertical, the distance C F is equal to p cos. < . 



ON THE ACTION OF PADDLE WHEELS. 



Every paddle wheel may be regarded as a series of levers, coming successively into action. 

 Each lever is represented by one of the arms of the wheels, at the extremity of which a float 

 is attached : the fulcrum is obtained by means of the reaction of the water on the paddle 

 board; the resistance to be overcome is that of the water to the motion of the vessel 

 acting at the centre of the wheel and in the direction of the vessel's course ; and the power 

 is that of the engine, applied at the extremity of the crank, which thus forms a bent lever 

 with each arm of the wheel. Now, since the fulcrum is obtained, as above stated, by means 

 of the reaction of the water on the float, and there can be no reaction unless the surface of 

 the float move through the water, it follows that the true fulcrum is situated at that point, on 

 which, if immersed, there would be no reaction. This is the point F found above ; for, as it 

 moves in the direction of the surface of the float, it can meet with no resistance from the 

 water. A certain expenditure of power is therefore necessary to force the floats through the 

 water, in addition to that required to propel the vessel. 



It is evident that the mean horizontal pressure on the floats of a steam vessel must be equal 



to the resistance opposed to the motion of the vessel ; therefore the 

 motive power must be equal to the mean product of the total pres- 

 sure on the floats by their velocity perpendicular to their surface, 

 plus the product of the mean horizontal pressure by the velocity of 

 the vessel. This sum in the common wheel is equal to the mean 

 product of the total pressure on the floats by their circumferential 

 velocity ; for let C A in the annexed figure represent a radius of 

 the wheel, C D a vertical line, and A a point of the surface of the 

 float ; also let A B, perpendicular to C A, represent the circum- 

 ferential velocity of the given point, and the horizontal line B G the 

 velocity of the vessel. Join A G, and draw G E parallel to 

 C A. The motion of the given point is in the direction of the 

 line A G, the length of which represents its velocity. Let p be 

 the pressure on the point A in the direction B A, perpendicular to the surface of the float ; 

 the pressure in the direction G A will be p sin. AGE, and the power required to overcome 

 that resistance at the velocity A G will be p sin. AGE x A G = p xAE. The horizontal 



T> -p 



pressure is equal to p cos. A B G = p x -p, which, multiplied into the velocity of the 



