ON THE CENTRE OF PRESSURE OF A PADDLE. 45 



and points without, in curtate or contracted cycloids. In Fig. 5. is represented the posi- 

 tion 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 in practice. 



It will be readily seen, that the effect of the vessel being in motion will be to roll the circle 

 abed on the line ef, so that the inner edge of every paddle will move through the cycloid 

 g h i, whilst the extremity moves through the cycloid klmno, 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 would, however, lead 

 to a calculation of greater labour than the nature of the present investigation demands ; and 

 as the circumstances upon which such calculations would be founded vary in every experi- 

 ment, according to the diameter of the wheel, depth of immersion, 8cc., two points have been 

 assumed, one of which is intended to meet the ordinary cases of slightly immersed, and the 

 other that of deeply immersed paddles. It appears, again, referring to the figure, that whilst 

 the extremity of the paddle is moving through the part of the curtate cycloid below the level 

 of the water, a point P in the radius of the wheel, which is situated in the circumference of 

 the rolling circle, has scarcely moved in the simple cycloid apr. The difference of the 

 curves during the lower part of the motion amounts nearly to what is due to an arc described 

 with a radius equal to the difference of the extreme radius of the wheel, and that of the circle 

 of equal velocity with the ship. 



It therefore appears, that the resistance on any part of the float varies nearly as the square 

 of its distance from the rolling circle ; and having at the same time taken into consideration 

 the greater length of time of the action of the extremity than of the inner edge of the paddle, 

 I find, from the examination of several experiments, that in the case of slight immersion 

 the assumption of the resistance on any point varying as the cube of the distance from the 

 rolling circle, and in deep immersions as the 2'5 power, will approximate very nearly to 

 the truth. 



Having thus assumed the law of resistance with respect to the radius, we readily find the 

 position of the centre of pressure by the following equation. 



Let r be the difference of the radius of the rolling circle and that of the wheel, n the power 

 of the resistance in relation to the radius, b the depth of the paddle, x any variable distance 

 from its upper edge, and y the distance of the mean centre of pressure, also from the upper 

 edge ; then the integral of (r + x) n d x will be the sum of all the resistances, and (r + y) b 

 the expression to which it is to be equal. 



We have therefore, when x = b, 



which, when = 3, gives 



and, when n=2'5, 



(2 (r + 6)3 \4 

 -TI ) - * 



