THE MOTION OF STEAM VESSELS. 315 



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 experiment, according to the diameter of the wheel, 

 depth of immersion, &c., I have contented myself with assuming two points, 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, C, in the radius of the wheel, which is situated in the 

 circumference of the rolling circle, has scarcely moved in the simple cycloid N C O. 

 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. 



I have considered from this cause 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 experi- 

 ments, that in the case of slight immersions the assumption of the resistance on any 

 point varying as the cube of the distance from the rolling circle, and in deep immer- 

 sions as the 2*5 power, will be a sufficiently near approximation for the present pur- 

 pose. 



Having thus assumed the ratio 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, h 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^ d x, will be the 

 sum of all the resistances, and (r -|- yY h the expression to which it is to be equal. 

 We have therefore, when a? = &, 



1 n4-\ n 



-^^ir^b) = {r+y) b, 

 which, when « = 3, gives 



^ = \-Tr) -^• 



And when r = 2*5, 



From these equations, the diameters to the centre of pressure of the common wheel 

 (given in column 16 of the following Table) have been calculated. 



In the new wheel, the centre of pressure will be nearly in the centre of gravity 

 when the paddle is totally immersed, the motion of the paddle being nearly vertical ; 



