Principle of the Screw to the Floats of Paddle-wheels. 253 



angle which P makes with the plane GCDH (the plane of 



x z) = -, the surface being a conoid, of which the helix is the 



directrix, and which is inclined at an angle of 45° to the plane 

 of xy } at a distance r from the axis. Hence 



z \ • z 



#=pcos- and y = psin — 



The inclination of the normal at any point to the axis of z is 



( s A 



I ,r cos- + ?/sm -I 



\f r 2 -f(#cos- + ?/sm-) 



which is equal to cos -1 . f - = 7, suppose. 



Vr + f 



Hence an element of the surface at the point P is 



p / 



The inclination of the normal to its trace on the plane of x y is 



r 



COS" 



vV + p 2 



This trace is perpendicular to P in the plane P I (fig. 3), 

 and therefore makes an angle P I with the direction of motion 

 of the point P. 



Let AC = OI = «, and let the inclination of P (-) to the 



plane of xz be denoted by } and let the angle P I = cj> ; then 

 the angle OP l = 0-<f> 



, n IN x(a+x) -fz/ 2 p + acos0 



cos (0 — 6) = , , — =—7== 



v r Vafi + y*S{a + aip + y* */ (a + xf + y* 



a being the inclination of the normal to the direction of motion 



of P, we get 



i a ,v r p + a cos 



cosa=sm7COS [u — ©)= — — = v — r — . 



Also, /3 being the inclination of the normal to the axis of y (the 

 direction of the required resistance), 



a - n rcosO 



cos p = sin 7 cos 0= . = • 



v r 2 + p 2 



Also, if co be the angular velocity of the wheel, then the velocity 



of the point P = ©xIP=G>x v /( a .f #) 2 + y 2 . 



