314 MR. p. W. BARLOW ON THE LAWS WHICH GOVERN 



To find the exact position of this point is, however, a question of very intricate 

 calculation ; and as it varies according to the depth of immersion of the paddle oi 

 float, the diameter of the wheel, and other circumstances, which vary in different 

 boats, I have contented myself with assuming a point which will meet the ordinary 

 cases, and which I have decided upon from the following considerations. 



It is very evident that in every case the resistance upon different parts of the 

 paddle is as the square of the distance from the centre of motion, because the resist- 

 ance of a fluid varies as the square of the velocity : this ratio is, however, always 

 increased more or less, in consequence of the extremity acting for a greater length of 

 time than the inner part. 



In the case of a wheel in motion, in a vessel at rest, if the length of the arc de- 

 scribed by the outer extremity of the paddle exceed that described by the inner edge, 

 in the ratio of the large radius to the smaller, the resistance upon any part of the 

 paddle will vary exactly as the square of the radius ; but this can only occur when 

 the wheel is either totally immersed or up to the centre of motion : in every other 

 circumstance it is evident that the arc described by the extremity will exceed that of 

 the inner edge in a greater ratio, depending upon the degree of immersion, radius of 

 wheel, &c. Consequently, the resistance upon any part of the paddle will increase in 

 a greater ratio than the square of the distance from the centre of motion. It is, more- 

 over, evident that the position of the centre of pressure will not only vary with every 

 change of immersion, but will continue to ascend from the moment the paddle enters 

 the water until it is immersed below the surface, when it becomes constant, and con- 

 tinues so until the upper part of the paddle again leaves the water. 



As these experiments are made entirely with vessels in motion, it is not necessary 

 to enter into a calculation of this precise point. I have merely spoken of the above 

 case with a view to facilitate the investigation of the more complicated question of 

 the centre of pressure of the paddle when the vessel is in motion. 



In this case it will be seen, that as the revolution of the paddle resembles a circle 

 rolling on a plane, every part of it will describe a cycloid. That point whose velocity 

 is equal to that of the vessel will move through a simple cycloid, points within that 

 circle in prolate cycloids, and every point without in curtate or contracted cycloids. 

 In fig. 2. is represented the position of the float of a paddle-wheel in different parts 

 of its revolution. The circumference, whose velocity is equal to that of the vessel, is 

 here equal to two thirds of that which passes through the extremity of the paddle, 

 which is about a medium case. It will be readily seen that the effect of the vessel 

 being in motion will be to roll the circle A B C D on the line E F, so that the inner 

 edge of every paddle will move through the cycloid R S T, whilst the extremity 

 moves through the cycloid L K H I M, as shown by the dotted lines in the figure. 

 As the centre of pressure varies at every angle of the paddle, in order to come at the 

 true position it becomes necessary to find the relative velocity of the two extremes of 

 the floats, or the distance moved in the two cycloids, at every instant of time. This 



