252 



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



density of the fluid, k the area, and v the velocity of that por- 

 tion of the surface. The effect of this resistance in a direction 

 inclined at an angle /3 to the normal, will be measured by 

 ^/cp^v 2 cos 2 a cos /3. 



Suppose A B G H to have made half a revolution round A B 

 from left to right, so that the „ 



floats are in their lowest posi- 

 tion, then the direction of mo- 

 tion of any point of the surface 

 being perpendicular to the line 

 drawn from it perpendicular to 

 the axis, we must find the angle 

 between the direction of motion 

 and the perpendicular to the 

 surface, and also between the 

 perpendicular to the surface at 

 that point, and the perpendi- 

 cular to the plane A B H G, 

 since the motion of the vessel is 

 perpendicular to this plane. If 

 a be the former angle and ft 

 the latter, then the resistance 

 B = ^rcpyV 2 x cos 2 « cos/3. Also 

 the power of engine required 

 will be measured by that part of the resistance perpendicular 

 to the surface (^fcpp 2 cos 2 a), which acts in a plane perpendicular 

 to the axis AB, multiplied by its distance from that axis — in other 

 words, the moment of the resistance about the axis of revolution. 



To express these inclinations and portions of surface mathe- 

 matically, we must have recourse to the ordinary notation of 

 Geometry of Three Dimensions. 



The surface of this screw is traced out by a straight rod, as P, 

 which slides uniformly along a fixed rod, as C D, at right 

 angles to it, and at the same time twists uniformly about that 

 fixed rod. It will be familiar to all as the surface formed by 

 the edges of the stairs in a spiral staircase, and the steepness of 

 the stairs will depend on the relation between the sliding and 

 the twisting motion, and also on the distance from the shaft. 



If the fixed rod C D be taken as the axis of z> and C G as the 

 axis of x, then the equation to the surface C G E K is 



. z z 



#sin- — ?/cos-=0, 



where, in this particular case, C G=r, since the angle between 

 G K and H G at G is 45°. If P, the distance of any point 

 P from the axis of the surface =p, then p 2 =# 2 +2/ 2 , and the 



