410 .Mir, SylvesUr on the Motions produced by the [Jut^e, 



I medium of any density, the velocity has a maximum at which it 

 urrives when the resistance of the medium is equal to the force 

 of gravity. After this, the body moves with a uniform velocity. 

 The time in which the body attains this maximum is as the ratio 

 between the surface and weight of the body, which may be a 

 ratio of any assignable magnitude, limited only by the extent to 

 which matter is divisible. This will be obvious when it is known 

 4hat jn so dividing matter, the weight decreases as the cube of 

 the diameter; while the surface diminishes only as the square 

 of the same. Hence matter of the greatest specific gravity may 

 be so divided that it may acquire a maximum velocity by I'alling 

 through the least possible space, acquiring a uniform velocity of 

 any assignable value. A body, whatever may be its figure, by 

 falling in a fluid, will describe a prism, the base of which will be 

 the surface which moves against the fluid. When the prismatic 

 column of the fluid, formed by the falling body, becomes equal to 

 the weight of the body, the resistance of the fluid will be equal 

 to the force of gravity, and consequently the body will, from that 

 point, move with a uniform velocity. If the surface presented 

 to the fluid during its motion be a hemisphere, it will meet with 

 only half the resistance which would be given to a plane surface. 

 Let a equal the surface presented to the fluid. 



h = the height it will fall from to acquire a uniform velo- 

 city. 

 = the specific gravity of the fluid. 

 = the specific gravity of the body. 



/ the length of the solid prism. 



Then/: B :: ?: A .*. Z ='^, and this is the same whatever 



may be the value of cr. 



When B is less than f, then the body will rise in the fluid, and 

 will attain its greatest velocity, when by its rising, it has formed 

 a prism equal in weight to its own bulk of the fluid, having its 

 own area of base and specific gravity. In this case, / B =/ A, 

 and / is greater than //, in the ratio that /'is greater than B. 



What will be the length of a prism of lead in order that it may 

 acquire its greatest velocity by faUing through one foot of 

 water? 



In the theorem / = ~, A = 12 inches, B = 11, and/= 1 .\ 



i 



B 



xl2 



/ = —rr- = It inches, the ength of the prism. If/ = —^ 



being the specific gravity of air, the value of I will be obtained 



for a prism faUing in the atmosphere, in which case / = — ^ of 



an inch whatever may be the area of the surface. If the falling 

 body be a sphere, then it will meet with only half the resistance 



