254 Mr. W. G. Adams on the Application of the 



Hence R = \p t v q K cos 2 a cos 



-2Pi<»r p (r 2 + p 2 ) Vi 



or, taking polar coordinates, 



_, s , (p -fa cos 0) 2 cos 0, ,« 

 K = fatA* . s _^ dp dO. 



Again, the perpendicular distance from the axis A B on the 

 trace of the normal in the plane of xy is 



p + a cos 0. 

 Therefore M, the moment about the axis A B 

 = ip^K cos 2 a sin 7(p + a cos 0) 



Each float extends to a distance (a + r) from the axis A B, 

 i. e. to the circumference of the wheel. Therefore for the limits 

 of p a 2 + p 2 + 20pcos0=(tf + r) 2 , 



or p + a cos = vV 2 + 2ar + a 2 cos 2 0. 



The limits for p are 



\/r 2 + 2ar + a 2 cos 2 — a cos and 0. 



The limits of integration for will be one- third of the ratio 

 of the breadth of the wheel to the length of C G. 



In the example worked out, C G has been taken = | A G, and 

 the breadth of the wheel G H has been taken slightly greater 

 than C G for convenience ; so tha*t the limits for are rather 

 more than ^ or 20°, and 0, and the breadth of the wheel is to 

 its radius as 7 : 10. 



The expressions for the resistance and moment about the axis 

 admit of integration with respect to p without any restriction : 



JfteoV 



j j ip+a ™l e £ cos0 dpdd 





(p* + r*) cos e + 2apco£ i e+ (a? cos 2 6-r*) cos0 , . 

 ^+p^ d P db 



a cos 2 6 . -^-^ dp dd 



F 



cos 2 — r 2 ) cos0 7 7/1 "l 



r *+ P * — d P de i 



