118 Mb. stokes, on SOME CASES OF FLUID MOTION. 



where F and Z are each Laplace's coefficients of the «'" order, and do not contain r. Let 

 f(9 w) "be the normal velocity of the point of the inner surface corresponding to G and w, 

 F(e, «0 the corresponding quantity for the outer; then the conditions which is to satisfy 

 are that 



-£ =/(0, to) when r = a, 

 dr 



^= F(9, w) when r = 6. 

 dr 



Let f(0, w), expanded in a series of Laplace's coefficients, be 



P, + P,... + P„+ ... 



which expansion may be performed by the usual formula, if not by inspection : then the first 



condition gives 



S: (nY„a'-' - (» + i)Z„a-<»^'') = ^^ P„; 



and equating Laplace's coefficients of the same order, we get 



nY„a'-' - (« + l)Z„a-<"-^''= P, (11). 



Let F(9, m), expanded in a series of Laplace's coefficients, be 



P'o + P', ...P'„+...; 

 then from tlie second condition, we get 



wF„6"-'-(n+l)Z„6-'" + ^'= P\, (12). 



From (11) and (12) we easily get 



P'b"^' - P^a"*' 



y = — 



(6^°+' - a'»+') ' 



ffl-"+' 6^''+'|f „&-'"-" -P„g-'°"''} 

 ^"= (n + l)(6«" + '-a-^" + ') ' 



provided n be greater than 0. If m = 0, we have 



-«-=Z„=P„, -6-=Z„ = P',, 

 But the condition that the volume of the fluid be not altered, gives 



a'fjf''f(e, io) sin OdOdco = h'f" pF[e, w) sin OdBdw, 

 or 4n-fl!=P„ = iT:WP\, 

 which reduces the two equations just given to one. 

 We have then, omitting the constant I'o, 



^ = - :^ + X {'''"■" - o'"-''}'' {- {P',.b"*' - P,.ri"'^)r" 



,,2n + ll2n+l 1 



+ - (P'„6-<'-"-P„a-<'' -"))•-<" ^'» (13), 



n + \ ) 



whicli determines the motion. 



When the fluid is infinitely extended, we have P' „ = since the velocity vanishes at an 

 infinite distance, and 6 = 05 , whence 



Pod' ^^ a''^'Pn 

 ^ r "' (n + !)?•"+' ■ 



It may be proved, precisely as was done, (Art. 8), for motion in two dimensions, that if 

 any portion of an infinitely extended fluid be disturbed by the motion of solid l)odies, or other- 



