OF FLUIDS ON THE MOTION OF PENDULUMS. 



[51] 



42. Let us now return to the case of the uniform motion of a sphere. In order to obtain 

 directly the expression for the resistance of the fluid, it would be requisite first to find p, then to 

 get P r and T 9 from (46), or at least to get the values of these functions for r = a, and lastly to 

 substitute in (47) and perform the integration. We should obtain p by integrating the expression 

 for dp got from (16) and (17). It would be requisite first to express u and q in terms of \j/, then 

 to transform the expression for dp so as to involve polar co-ordinates, and then substitute for 

 yf/ its value given by (121); or 'else to express the right-hand member of (121) by the co-ordi- 

 nates », •&, and substitute in the expression for dp*. We have seen, however, that the results 

 applicable to uniform motion may be deduced as limiting cases of those which relate to 

 oscillatory motion, and consequently, we may make use of the expression for F already worked 

 out. Writing V for ce^~ int in the first equation of Art. 20, expressing m in terms of n, 

 and then making n vanish, we get 



- F=6iriu'paV, (126) 



and — F is the resistance required. 



This equation may be employed to determine the terminal velocity of a sphere ascending 

 or descending in a fluid, provided the motion be so slow that the square of the velocity 

 may be neglected. It has been shewn experimentally by Coulomb j-, that in the case of very 

 slow motions, the resistance of a fluid depends partly on the square and partly on the first 

 power of the velocity. The formula (126) determines, in the particular case of a sphere, that 

 part of the whole resistance which depends on the first power of the velocity, even though the 

 part which depends on the square of the velocity be not wholly insensible. 



It is particularly to be remarked, that according to the formula (126), the resistance varies 

 not as the surface but as the radius of the sphere, and consequently the quotient of the resist- 

 ance divided by the mass increases in a higher ratio, as the radius diminishes, than if the 

 resistance varied as the surface. Accordingly, fine powders remain nearly suspended in a 

 fluid of widely different specific gravity. 



43. When the motion is so slow that the part of the resistance which depends on the 

 square of the velocity may be neglected, we have, supposing V to be the terminal velocity, 



* The equations(16), (17) give, after a troublesome trans- 

 formation to polar co-ordinates, 



dp 



dr r 1 sin 8 



— If- 



dd \J? 



sin d 1 d o d\ 



-1 *, (a) 



dp 



dJ' 



d8 sinfl dd fi dt 



a d_ /d>_ smjj jj 1 d p d \ 

 ~sinfl dr \dr' + r' rf0 sin? dl ~~p dl) *' * <• 



The expression for dp got from these equations is an exact 

 differential by virtue of the equation which determines \\r ; and 

 in the problems considered in Section IX and in the present 

 Section \jr has the form ¥sin»0, where ¥ is independent of 0. 

 Hence we get from (4), by integrating partially with respect 

 to 0, 



. d /d° 2 p d\ 

 ^^^d-rKdT'-^-^Tt) 



*• (c) 



1 £* 



dr \dr 2 r 3 << dt) 



It is unnecessary to add an arbitrary function of r, because 

 if X(r) be such a function which we suppose added to the 

 right-hand member of (c), we must determine X by substituting 

 in (o). The resulting expression for X'(r) cannot contain 0, 

 inasmuch as the expression for dp is an exact differential, but 

 it is composed of terms which all involve cos as a factor, and 

 therefore we know, without working out, that these terms must 

 destroy one another. Hence X (r) must be constant, or at most 

 be a function of t, which we may suppose included in TJ. X (r) 

 will in fact be equal to zero if II be the equilibrium pressure at 

 the depth at which fgdz' vanishes. 



t Mimoires de VInstxtut, Tom. m. p. 246. 



31—2 



