as Floats for Paddle-wheels. 353 



Suppose that a portion of the screw-blade nearest the axis of 

 breadth kr is cut away, then this will be the lower limit for p in 

 the integration. 



If only that part of the float were cut away which tends to 

 stop the boat, then the lower limit of integration would be given 

 by the equation <o(p -j- a cos 0) — v cos 0=0, or 



= ( )r eos0. 



\o)r r/ 



Or if the part cut away be of the same width throughout, then 



( v a\ 



or 



\cor r) 



T 



Supposing, as before, that a=— } then 



& 



£=*(!+»*)• 



a v 



Without these suppositions the quantities k, -, and — are in- 

 dependent of one another ; and it may be considered doubtful 

 whether (1) they should have the same values for wheels of dif- 

 ferent radii, or (2) whether the above are the best hypotheses 

 which can be made. It is only by repeating the calculations for 

 different cases that these points can be theoretically determined. 



Taking the above values of - and — , the resistance on both 

 floats becomes 



ip,a>V { M (cos 0d0 + log e (l + Jc 2 ) f cos 2 d0 



- 2 tan- 1 k ( cos 0d0 + -J- tan" 1 k \ cos 3 d0 



- (1 + 2k) log e (1 + B) f cos 2 d0 

 -itan- 1 A;(cos 3 ^^ + 2F.tan- 1 ^(cos 3 6>^} 



