392 Hydrodynamics. 



in the duplicate ratio of the radius to the sine of the inclination, 

 the resisting force m would be 



71 S D 2 V 2 i 2 71 S R 2 V 2 i 2 



But if the body were terminated by an end or surface of any 

 other form, as a spherical one, where every part of it has a dif- 

 ferent inclination to the axis ; then a further investigation be- 

 comes necessary. 



503. To determine the resistance of a fluid to any body moving 

 in it, having a curved end as a sphere, a cylinder with a hemispheri- 

 cal end, fyc. 



Fig. 241. 1. Let BEAD be a section through the axis CA of the solid, 

 moving in the direction of that axis. To any point of the curve 

 draw the tangent EG, meeting the axis produced in G ; also draw 

 the perpendicular ordinates EF, e/, indefinitely near to each 

 other ; and draw a e parallel to CG. 



Putting CF = a?, EF = t/, BE = z, t = sine of the angle G, radius 

 being 1 ; then 2 ny is the circumference whose radius is EF, or the 

 circumference described by the point E, in revolving about the 

 axis CA; and 2^t/ x EC, or %7iy d z, is the differential of the 

 surface, or ir is the surface described by E c, in its revolution 

 about CA ; hence 



$v 2 i* ^ j 7i s v 2 i 3 



X Znydz, or - X ydz 



is the resistance on that ring, or the differential of the resistance 

 to the body, whatever the figure of it may be ; the integral of 

 which will be the resistance required. 



(2.) In the case of a spherical shape ; putting the radius CA 

 or CB = R, we have 



/ o ^ EF CP x 



J, = V(R.. -*), = - = - = -, 



and ?/eZzorEF X Ee = cEX ae = ndx-, 



therefore the general differential 



3TS W 2 



