Mr. Tredgold on the Theory of Hydro-dynamics. 115 



Prop. 7. — The pressure of a fluid in a direction perpendi- 

 cular to the surface which it follows, is as the difference be- 

 tween the height of a column equal to the pressure of the fluid, 

 and the height of a column of fluid capable of generating the 

 velocity of the surface. 



Let m be the height of a column equal to the pressure of 

 the fluid on a surface at rest, and h= the height of the column 

 capable of generating the velocity ; then m — h == the height 

 of the column which generates the effective motion, for what- 

 ever pressure is lost by the motion of the surface must be sub- 

 tracted. 



Prop. 8. — The impelling force of the fluid which follows a 

 body in motion, in the direction of the motion, is equal to a 



-, r r% • i i 1 • i . • 64 m r 2 sin a — w- sin 3 a 



column ot fluid whose height is . 



Let v be the velocity of the body, then the velocity with 

 which the fluid flows in a direction perpendicular to the sur- 

 face is , (Prop. S.) and the head h equivalent to this 



velocity is ■ 64 r3 — ; consequently, (Prop. 7) til — h = m — 



v" 2 sin - a 64 m r - — v - sin 2 a 1 i • t -, ■, 



6 , r „ — gj-^ ; and, tins pressure reduced to the 



j- ,• p.-i .' • 64 mr- sin a — v- sin 3 a , _, n 



direction of the motion is i = the effect of 



64 r 3 



the fluid which follows the body. 



Cor. The anterior and posterior forms of the body should 

 not be similar figures in the solid of least resistance. 



Prop. 9. — The total resistance of a fluid to a body moving 

 in it is equal to the pressure of a column of the fluid of which 



.1 i • i . • 3 v n - sin s a — 64 m r 2 sin a ■. , , . , 



the height is ■ — -— , when the ends are iden- 

 tical figures. 



We have found the direct resistance to be v = - v sirt ? > 



32 ri 64 r» 



from whence if we deduct the effect of the fluid which follows 



., i 2i>- sin 3 a — 64#tr 2 sin a -4- v- sin 3 a -, . , 



it, we have g— ; and sin a in both 



parts of the expression being made of the same value from 



•j ,-, en i 3 v- sin 3 a — 64 m r- sin a , 



identity ot figure, we have ■ — - — = the co- 



04 r 3 



lumn equivalent to the whole effect. 



Hence we see that the resistance is not as the square of the 

 velocity ; and besides, this equation shows the important cir- 

 cumstance, that when the velocity exceeds that which the me- 

 dium is capable of propagating, the fluid must accumulate be- 

 fore the body, till its density, or modulus of elasticity, becomes 



P 2 sufficient 



