of Attractive Forces. 331 



periodic. Hence, putting V for the velocity perpendicular to r, 



= 3- sin (Ar + c') sin ^. 5;. m sin (nt + e). 



Consequently, by integration, 



^ sin (/:r + c') . ^ hmb'^ . ^ ^ 

 V = -3 >- sin ^ . S . -;p-cos {nt^-e), 



the velocity being supposed to vanish where r is indefinitely 



great. Similarly, if U be the velocity in the direction of ?•, it will 



be found that 



^. 2sin(/vr + c') . ^hmm , ^ , 

 U= ^3 ^cos^.S cos(H^ + e). 



To employ these equations for finding expressions for the velo- 

 city of fluid compelled to move in contact with the surface of a 

 sphere, as supposed in the foregoing problems, it is only neces- 

 sary to suppose r to be equal to the radius of the sphere. We 

 thus get values of V and U identical with those resulting from 

 the solutions of these problems, which values are thus proved to 

 be consistent with the general hydrodynamical equations. 



It may here be remarked that the foregoing value of U varies 

 inversely as the cube of r, whereas it has hitherto been assumed 

 that the velocity in the direction of a radius produced varies in- 

 versely as the square of r. To explain this apparent discrepancy, 

 it is to be observed that the velocity obtained by the preceding 

 investigation is exclusively that which is accompanied by con- 

 densation. This will be shown hereafter : at present it suffices 

 to state that the above values of V and U make Nrd^-\-^5dr an 

 exact differential. 



There is also another supposition by which a definite solution 

 of the equation (y) may be obtained. Let ■^=-^[r) cos 0. Then 

 the equation becomes 



(7^-^ + ^'%)sin^cos^ = 0, 



which being integrated, gives with sufficient approximation, 



A' 

 X= -jaSin (A;/- + c"). 



From this value may be obtained, by proceeding as above, 



^, sin(A:r + c") • ^. „ Ihi^b'' 



V '= 2^14 — '- sin 2^ . S . — ^ cos {nt + e'), 



,,,_ 3 sin (Ar + c") „. „ AWA^ 



^ = ^^ cos 2^|. S . cos {nt + e'). 



