44 APPENDIX. 



With the aid of these experiments, and the data given with them, I have been enabled to 

 calculate in each case the relative velocity of the wheel and vessel, the actual pressure upon 

 the vertical paddle, the area of paddle per horse power, and other facts; by the comparison of 

 which with the velocity, tonnage, &c., a just conception may be formed, not only of the action 

 of each paddle wheel, and the manner in which the power of the engine is expended, but also 

 the amount of surface, proportion of the paddles, dip of immersion, &c., which are required 

 to enable an engine to produce the greatest useful effect. 



The manner of obtaining these calculated results will be at once evident from the heads of 

 the columns, except the last five, which relate to the velocity of the vessel as compared to the 

 wheel, and the amount of pressure exerted by the engine on the vertical paddle. 



As the resistance opposed to the paddle is distributed over a considerable depth of paddle 

 board, it is necessary, in comparing the relative velocity of the wheel and vessel, to find that 

 point in which, if all the resistances were concentrated, they would have equal effect in pro- 

 pelling the vessel : this point is termed the centre of pressure, and its exact position becomes 

 a question of very intricate calculation, as it changes according to the depth of immersion, the 

 diameter of the wheel, and other circumstances, which vary in different boats. It was 

 necessary therefore to assume a point which would meet the ordinary cases; and this has 

 been decided upon from the following considerations. 



ON THE CENTRE OF PRESSURE OF A PADDLE. 



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 resistance 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 described by 

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

 larger radius to the smaller, the resistance upon any part of the paddle would 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, See. ; 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 moreover 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 

 continues 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 ; the above case being alluded to merely with a view of 

 facilitating the investigation of the more complicated question of the centre of pressure of the 

 paddle when the vessel is in motion. 



Here 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 rotary velocity is equal to that of 

 the vessel will move through a simple cycloid ; points within that circle, in prolate cycloids ; 



