Mr. stokes, on SOME CASES OF FLUID MOTION. 115 



~ logrf"f{ff)dff - -Sr - (-)" Tne') COS 71 (9 - &)dff. 



which becomes on summing the series 



^ \ogr ry{9')d9' + - r''log (i - 2 - cos (9 - &) + ^l Vcev^*' ; 



•iTT •'0 TT -"o ( r r j 



dd> a fi" (I ar cos (9 - 9') - a- 1 



whence ■—£- = — / {- + — ^^ '—, > f(9)d9 



dr -KrJ, X-i 1^- Marcos 1,9-9) + d^y^' 



If we suppose r to become equal to a the quantity under the integral sign vanislies, except 

 for values of 9', which are indefinitely near to 9. The value of the integral itself becomes /(0)*. 

 Hence it appears, that to the disturbance of each element of the surface, there corresponds 

 a normal velocity of the particles in contact with the surface, which is zero, except just about 

 the disturbed clement. The whole disturbance of the fluid will be the aggregate of the dis- 

 turbances due to those of the several elements of the surface. The case of the initial motion 

 of fluid within a cylinder, and the analogous cases of motion within and without a sphere, which 

 will be given in the next article, may be treated in the same manner. 



The velocity in the direction of r given by equation (7), ( = — ^| , 



= 1-2,, - \C„cosnd + D„h\nn9\, 



r \r J 



and that perpendicular to r, and reckoned positive in the same direction as 9, ( = — ~] , 



\ rd9j 



= ^' i-X {C„s\nn9 - D„cosn9\. 



\r I 



Conceive a mass of fluid comprised between two infinite parallel planes, and suppose that 



a certain portion of this fluid contains solid bodies bounded by cylindrical surfaces perpendicular 



to these ])lanes. The whole being at first at rest, suppose that the surfaces of these solids are 



moved in any manner, the motion being in two dimensions. Conceive a circular cylindrical 



surface described perpendicular to the parallel planes, and with a radius so large that all the 



solids are comprised within it. Then, (Art. 4.), we may suppose the motion of the fluid at any 



time to liave been produced directly by impact. On this supposition the initial motion of the 



part of the fluid without the above cylindrical surface will be determined in terms of the normal 



motion of the fluid forming tliat surface, as has just been done. If C„ be different from zero. 



aC 

 then, at a great distance in the fluid, the velocity will be ultimately — "- , and directed to or from 



)• 



the axis of the cylinder, and alike in all directions. Since the rate of increase of volume of a 



length / of the cylinder is equal to la jj f{9')d9' = ZirlaCa, it appears that the velocity at 



a great distance is proportional to the expansion or contraction of a unit of length of the solids. 



If however there should be no expansion or contraction, or if the expansion of some of the solids 



should make uj) for the contraction of the rest, then in general the most important part of the 



C' cos 6 

 motion at a great distance will consist of a velocity z '- directed to or from the centre, and 



C'sin^i 

 another — , — perpendicular to the radius vector, the value of C and the direction from wiiicli 



9, is measured varying from one instant to another. The resultant of these velocities will vary 

 inversely as the square of the distance. 



• PoiftSon, T/itorie dc la Chaleur, Chnp. vii. 



P 2 



