SECT, x.] OF STEAM NAVIGATION. 301 



sometimes made to slide on the arms, so as to reduce or increase the depth of 

 immersion in the water, according as the vessel is more or less laden. 



629. To determine their power, let V be the velocity of the exterior portion of 

 the wheel, and r its radius ; then the velocity at any distance r x from the centre is 



( r ~ x > . B u t while the paddle acts on the water, the vessel moves forward, and 



the water recedes only at the rate of the difference between the velocity of the 

 wheel and that of the vessel ; therefore, if v be the velocity of the boat, 



P^ v = the velocity with which the paddle strikes the water. Hence, as 



the quantity of water so struck and put in motion is nearly proportionate to 

 the velocity of the vessel, we have, 



1-5 v - = the resistance to one square foot of the paddle ; 



and, making b = the breadth of the paddle, 



powel , 



The integral is 



1-5 v 2 * (V r x | V ** r v a) . 



. . - _ -- - , 



r 

 and when the depth of the paddle is h, it is 



1-5 b hv* / (r 4A) V-rvj ,. 



_ \ v 2 ' __ / = the direct power, 



r 



which must be equal to the resistance of the vessel. 



The loss by oblique action has to be estimated before the power of the engine 

 can be found, but previously we may proceed to determine the best velocity for the 

 wheels in still water. 



630. Let the equation be freed of all its constant multipliers, except those 

 relating to or connected with its velocities ; then it is v z (V r % V h r v) ; and 

 making v variable, its differential is 2V r v d v V h v dv 3 r v 2 dv = o ; from 



whence we have, 



3 r 2 (r 4 A) , 



V = a^rrTT) and = -SrF- ' v - 



The excess of the velocity of the outer point of the paddle therefore depends in 

 part on its depth compared with the radius ; the greater the depth, the less excess 



is necessary. 1 



t 



1 The above is according to the principles of Tredgold's investigation, but it is not here 

 sanctioned as to its accuracy. Although the quantity of water put in motion may be proportional 

 to the velocity of the vessel, it does not appear that the whole of this water is struck by the paddle, 



