302 OF STEAM NAVIGATION. [SECT. x. 



631. If this value of V be inserted in the equation for the area of the 

 paddles, then -75 v 3 b h = the effect of the paddles, which must be equal to 



as is assumed at the outset ; on the contrary, a large portion is evidently put in motion by second- 

 ary pressure. The pressure on the paddle must in all cases be nearly proportional to the square 

 of its velocity (V v) through the water, V denoting the velocity of the " centre of pressure," 

 which is near to the centre of the paddle. It should however be observed, that the passage of 

 each paddle leaves a vacuum behind it, and that its repletion causes a temporary stream, which 

 necessarily diminishes the effect of the succeeding paddle. Consequently, if we represent the 

 pressure by c (V v) 2 the coefficient c must be rather less than in the case of the motion of a body 

 through still water ; and it can only be regarded as constant for quick velocities, as the diminu- 

 tion from the cause alluded to will be most considerable when the velocities of the vessel and 

 paddle are slow. 



If we suppose the vessel to be influenced only by the engine and the resistance of the water, 

 and disregard the variations of her immersion, the resistance will obviously be proportioned to v", 

 and, the velocities being uniform, it must be equal to the reaction or pressure against the paddles, 

 in the direction of the motion of the vessel. Let C v 2 denote this pressure, and let k be the distance 

 of the centre of pressure below the axis, or the leverage with which it acts ; then C k v 2 is the 

 effort to turn the wheel. As k t> 2 must therefore be constant under given circumstances, whatever 

 be the dimensions of the wheel and paddle, it is evident that the velocity will be increased by 

 diminishing k, and that in this respect small wheels are preferable to large ones, were it not for 

 the power required to overcome the increased oblique action, and the inconveniences attending it. 

 In strictness k refers to the centre of the entire horizontal pressure against all the paddles im- 

 mersed, and this will be near the middle of that which is most depressed. If we suppose k to be 

 proportional to r, the effort to turn the wheel may be represented by C' r v 3 , and, <u denoting the 

 angular velocity of rotation, C' r <> r 2 = C' V v 2 will then represent the whole power expended by 

 the engine. It hence also appears, that when the engine works with a given power and a given 

 speed, both r v- and w must be constant for all sized wheels, the dimensions of the paddle and 

 the radius of the wheel being properly adapted to each other, for the due consumption of the 

 steam generated in the boilers. 



To put the relations in an analytical form, let the notation be as follows : 



P the pressure on the piston in Ibs. per circular inch, 



a ,, radius of the cylinder, 



/ length of stroke, 



n number of strokes per minute, 



a velocity of the piston in feet per second, 



v velocity of the vessel ,, 



V ,, velocity of the centre of pressure of the wheels, 



k the radius of the wheel to the centre of pressure ; 



and let B 2 v- be the pressure or resistance, in Ibs., experienced at the bow of the vessel, and 

 i (V ) 2 that on the paddles, both estimated in the line of motion, so that 



B v = b (V - v) - - (1) 



then B 2 , 4 2 , will be factors proportional to the equivalent resisting surfaces of the boat and 

 the paddles immersed. And since, in respect of the engine, the pressure B 2 v- on the 



