ON PADDLE WHEELS. 125 



Let n be the number of revolutions of the wheel per minute, and p the radius of the rolling 



circle; the distance travelled by the vessel in a minute = 60 v = 2 TT p n, whence v = - . 



30 



In like manner V = - These values being substituted in the expression of the effective 



velocity, it becomes 



TT n r 



[at-pcoB. +]. 



The factor x p cos. <f> is the distance of the given point from the point F (V. Fig. page 120). 



Let b be the breadth of the float, then b.dx will be the area of an element of its surface 

 extending from one side to the other, and the resistance opposed to its motion, according to 

 the theory already laid down, will be equal to 



w being the weight of a cubic foot of water. 



If we multiply this expression into the circumferential velocity of the element of surface, we 

 have already shown that the product will be the power expended upon it, or the differential of 

 the power expended upon the surface of the float while in the given position. We have 

 therefore, calling^ this latter quantity, 



27000 x2y . 



and supposing the upper edge of the float not to be immersed below the surface of the water, 

 in which case the limits of the values of x are R and 



27000 x 



COS. <f> 



/R 

 [x p cos. </>] 2 x. dx 

 3 



cos. <t> 

 1 n 3 bw i 1 2 _ 1 la 4 



27000 x2 ff 



If we call P the mean power expended on the float during a whole revolution, on the 

 supposition that it is never immersed above its upper edge, and a. the angle at which it enters 

 and leaves the water, we shall have 



7r 2 w 3 6w " a 



p = _ / [3fl<-8#>cos.</> + 6flVcos.</> 2 - ~^~ + ^-^ - 6V] rf^. 



324000 x 2c/ J L cos.</> 4 cos.^> 2 



By integration we find, after simplifying, 



_ 7T 2 W 3 AW 



In cases where the upper edge of the float is immersed during a part of the stroke to a 

 certain depth below the surface of the water, the above value of P includes the power which 

 would be expended on a paddle-board extending from the upper edge of the actual float to 

 the surface of the water at the deepest immersion of the radius. This quantity, which is to 

 be deducted from the former, is found by substituting in the value of P, r for R, and /3 for , 



r 



