114 Mr. Tredgold on the Theory of Hydro- dynamics. 



Let v be the velocity of the plane, a the angle it makes with 

 the direction of the motion, and h the height of the column 

 equivalent to the pressure perpendicular to the plane. 



Then (Prop. 3.) r : sin a : : v : = the velocity in a di- 

 rection perpendicular to the surface; and since (by Prop. 4.) 



( V 



64 h i w 5 sin 2 a 7 

 ; we nave „„ , — = n. 



2 ' 32 r~ 



But the resistance in a direction perpendicular to the surface 

 is to the resistance in the direction of t the motion as rad : sin a ; 

 (Philosophical Magazine, vol. Ixviii. p. IS.) hence r:sin a :: 



v 2 sin 2 o i>2 sin s a 



——!L — the height of a column of the fluid equi- 



32 r'i 32 rl _ 



valent to the resistance. 



Cor. The weight of the column will be — — ■ - - . when 



c is the area of the base, and w the weight of a cubic foot cf 

 the fluid. 



Prop. 6. — The direct resistance of a solid having a circular 



. . fp iP-w y y sin 3 a 



baseisy jg- . 



Let j? = 3*1416, and y — the variable radius of the base; 



then c — 2pyy ; hence /^-y ^g = the fluent of the re- 



sistance. 



1. When the solid is a cylinder the plane surface of the 



end being perpendicular to the direction of the motion JL^lll 



= the resistance. 



2. If the solid terminate in a cone of which the slam^side is 



r, and the sin a = y, we nave — r^-r— = the resistance. 



a* 32 r 3 



When r" = 2 ?/ ; we have '" ■ I- = the resistance, whence 

 the direct resistance of a cylinder is to such a cone as 168 : 100. 



3. When the solid terminates in a hemisphere, r — x = 



■%pwv' i (r — x) l x' pwv q r~ 



sm «, and ?/?/ =r—xx; therefore /- — — - — - — = — 



'•■'•' J lo r 3 5 x 16 



the resistance : hence, the resistance of a hemisphere is to that 

 of a cylinder as 2 : 5 or as 10 : 25. 



Leaving these cases 1 will now return to the effect of the 

 fluid closing upon the body behind it. 



The velocity with which the fluid closes on the body it is 

 obvious should be estimated in a direction perpendicular to 

 the surfaces, and, following the same train of reasoning as, be- 

 fore, we shall find no difficulty in determining its effects. A 



Pi- 



