OF THE MOTION OF FLUIDS. 189 



SECTION III. 



Application of the Principles of the foregoing Section to an instance of 

 the Resistance of an Incompressible Fluid to a Body hounded by a 

 Spherical Surface moving in it. 



13. Let a solid sphere, partially immersed in water, being of less 

 specific gravity than the fluid, be drawn along in a horizontal direction 

 with a given uniform velocity ; it is required to find the height of 

 its centre above the horizontal surface of the water. 



We shall suppose for the sake of simplicity, that the fluid is 

 unlimited in extent both in the vertical and horizontal directions, and 

 that the surface of the sphere is so smooth that it impresses no velocity 

 on the water in contact with it in the direction of a tangent plane. 

 Let CDJBJE (Fig. 3.) be the sphere, O its centre, ADE the intersection 

 of the horizontal surface of the fluid by a vertical plane through the 

 centre of the ball; OQ a line through the centre parallel to ADE. 

 This will be the direction of the motion of O, since the velocity is 

 supposed to have become uniform, and ON to be constant. Let A, 

 a fixed point in ADE, be the origin of co-ordinates, AN=a, NO = 'y, 



at any instant. Then the velocity {V) of O = -r-. Draw OB vertical; 



let P be any point of the surface immersed; through P draw the 

 spherical arcs PQ, PB, and let the angle QOP=6, and the angle 

 PQB = to. The velocity impressed by the sphere on the fluid at P 

 is F'cos 9, as none is impressed in the direction of a tangent plane. 

 This velocity is directed to the point O, because in the case of a 

 spherical surface / = 0. Hence if « = the radius of the sphere, 



C 



FcosO = — . (Art. 11.) The velocity at every point of the line OP 



produced, wiU at a given instant be in the direction of this line, 

 Vol. V. Part II. B b 



