SECT. X.] 



OF STEAM NAVIGATION. 



303 



the resistance of the vessel ; and the comparison is easily made by means 

 of the equations for the resistance of vessels (art. 622). The power required 



paddles acts with the velocity V, and the pressure P a 2 on the piston acts with the velocity a, 

 we have, neglecting the effects of oblique action, 



P a* u = B 2 v" V - - - (2) 

 We have also, from the mechanical arrangement of the parts, 



V = 



k 



2nl 



"* ~60 



nl 



(4) 



These four equations, which embody the whole theory of paddle wheels, may be easily discussed 

 for any purpose that may be had in view. The following expressions, deduced from them, are the 

 most commonly applicable : 



v = _ 



b 

 B + 



= 



77"' k 3 



B b \ V' 



*_ } 



+ b) 



(B + ft) f 3 

 b Pa 2 



30 



() 



K) 



_L X^^ 



r = \ "B-V/ _ * 



V = 



n k 



* -30- 

 B + 6 



Pa 2 / 



* A 



- (V 2 ) 



- (V s ) 



Equations ( 3 ), (V 3 ) show that with given sized paddles and draught of water, the velocities, under 

 all circumstances, will be proportional to the cube root of the quantity of steam consumed per 

 minute. Also equation (tij) shows that more steam may be consumed by either increasing the 

 length of the stroke or diminishing the radius of the wheel, while (t> 5 ), (V 2 ) indicate a corresponding 

 increase of the velocities in the ratio of the cube root of the increased consumption. It is also 

 evident from these last equations, that if the paddle be lessened or b diminished, the velocities 

 u V and consumption of steam will be increased, while v will remain unaltered. 



For any particular vessel the constants B, 6, may be determined from a few good experiments ; 

 the former will be nearly proportional to the square root of the draught of water, and the latter 

 nearly as the square root of the area of one of the paddles. ED. 



