252 Mr. Tredgold on a new Theory of the Resistance 



experiment, it will be useful to consider the different cases 

 corresponding to the experiments to be compared. 



12. If a plane of given area, and equal thickness, be set at dif- 

 ferent angles to the direction of the motion, the resistance will 



be expressed by -^- j2(sin 2 a-{-cos 2 fl)+ sin a + cosfl? = H 



when the effect of friction is neglected. (See Plate IV. fig. 3.) 

 When the thickness is inconsiderable, the effect of the edge 



may be neglected, and then — (2sin 2 tf -f sin a) = H. 



13. In the case of a parallelepiped, with a wedge-formed 

 prow attached, we have sin c = 1 ; ^see fig. 2.) and, 



— (2sin 3 « + 1) = H. 

 4g v 



Including the friction £ (2 sin 3 a + 1 + ffift *?•»«) ) = H. 



The same equations will apply when a cylinder terminates at 

 one end in a cone. 



14. The resistances of curved surfaces may also be com- 

 puted: for example, let it be the resistance of a sphere; — put 

 p =• 3*14159, and y = the variable radius of the base, and 

 2pyy = the fluxion of its area. Then the figure being a circle, 

 and x the abscissa measured on the direction of the body's 



motion (fig. 4.) — — = sin a ; and yy = (r—x) x; consequently, 



; — *. — 3. -v 



2pv 2 l 2r — xx r—xx I « fl 



4rV— - + -7T-J=Hx2 W 

 The fluents are, 



2p«>«C g(rS-(r-« )0 H-(r-x)* y _ 



And, when x = r, 



4g \ 5 2 / 4g r 4g 



The resistance of a sphere is to the resistance of a cylinder 

 of the same diameter as 1*3 : 3, or as 1 : 2*308 ; or as *433 : 1. 



15. If a cylinder have a hemispherical end or prow, then 

 sin c becomes unity, and substituting this value of it in the 

 equation we have, 



16. And, if the motion be reversed, or the flat end go for- 

 ward, t* / 1 \ 2-5 «« _ 



17. Ac- 



