148 APPENDIX. 



Effect. veL = [R cos. < - a], 

 30 



The corresponding pressure on the float (which is all effective) is equal to the product of its 

 immersed area by the square of its effective velocity and the fraction . The immersed 



w t/ 



area, supposing the float to extend to the surface of the water, is equal to the product of its 

 dip, B' A', by its breadth b, and the dip is, in general, equal to R cos. <f> a ; therefore the 

 pressure is 



7T 2 M 2 W b r-f, -I, 



p = [R cos. A a] 3 . 



900 x 2 g L 



The moment of resistance due to the given float, being equal to the sum of the products of 

 p by the effect, vel. of the float, and by the velocity of the vessel respectively, is equal to 

 p [F G + G E'] p. F E' =p V cos. <f>, or substituting forp and V their values, 



n 3 R n 3 w b rr) -,,, 



R cos. <z> a 3 cos. <p. 



27000 x 2ff L 



The mean value of this expression for the whole revolution of the wheel is equal to the 

 definite integral 



a 

 n 2 R n 3 w b 



27000 x 2y 



o 

 which is 



/ [R cos. cf> a] 3 cos. (f>. d $, 



-([3 R 3 + 12 R a 2 ] a - [13 R? a + 2 a 3 ] sin. a ). 



216000 x 2 r\ 



This is on the supposition that the float extends to the surface of the water during the 

 whole stroke. If otherwise, let A' B' be the position of the float at the instant its upper 

 edge enters the water, and let the inclination S' C D = /3 ; we shall then have R cos. 

 /3 = a + f, which for simplicity we will call k. Let A" B" (see the figure) be one of the 

 positions of the float, in which its upper edge is immersed to a certain depth I B" below the 

 surface of the water. In this case we must deduct from the above resistance the portion due 

 to an area of float, whose length is equal to b, and breadth to I B". Now I B" = R cos. < k, 

 so that the required area is equal to b [R cos. <f> k], and the moment of resistance due to 

 this area is 



7T 3 Rn 3 W b rTS lorn 



\R cos. <6 a] 2 |K cos. <p k] cos. <t>. 



27000 x 2ff L 



Its mean value is equal to the definite integral 



^ 

 Tr 2 R n 3 w b 



/f-R cos. <4 a] 2 T.R cos. rf> k~\ cos. <f>. ? <f>, 

 L V J L V J V V, 



27000 x 2^r 



o 



which, after integrating and substituting for cos. /3 its value , becomes 



R 



HT* ft n^ in hi \ 



( [9 # 3 + 12 7? 2 + 24 .R a /t] ^- [32 R* a + 7 fl 2 k+ 12 a 2 /t-8 a F + 2 P] sin. ft). 

 648000 x 2 y\ L I 



