136 Mr. stokes, ON SOME CASES OF FLUID MOTION. 



Employing the same notation as in the last case, let F {.v, y) be the given normal velocity at any 

 point of the face in the plane ay. Let fo fo J^ (.-v, y) dwdy = Wal, and let 



F{.w,y)^f{cc,y) + W : 



then, since the normal motion of the above face due to the function / (d-, y) does not alter the 



volume of the fluid, we may consider separately the part of cp due to this quantity. For this 



part we have 



d'd) d'd) d'd) , ^ 



—J- + — i + — i =0 (4.1), 



dx* df dx' ^ ' 



with the conditions 



—^ = 0, when a' = or o (12), 



-2 = 0, when ?/ = or t (43), 



dy 



— i- = 0, when X = c (44), 



dz 



^=/C^', !/)> when^ = (45), 



within limits corresponding to those of the box. 



For a given value of x the value of (p from >t = to .r = n and from y = to y = b may be 

 expanded in a series of the form 



2o S„ P,„.„ cos cos — -- , 



a b 



the sign 2 referring to m, and 2' to n: and since the values of d>, —^ and — -^ do not alter 

 ° ^ dx dy 



abruptly, and equations (42) and (43) are satisfied, it follows that the series by which <p, 



— and — are expressed are convergent, and hold good for the limiting values of .i- and y. 

 dx dy 



Substituting the value of <p just given in (41), equating to zero coefficients of corresponding 



cosines, and introducing the condition (44), we have, omitting the constant, or supposing A„a = 0, 



(p = 2,0 \ .^,»,„j£ "■ +e ' i cos cos — - — , 



, p' iif of 

 where — = — +—. 

 c a- 



Determining the coefficients such as J,„„ from the condition (45) in the usual manner we have, 

 m and n being > 0, 



."in ri ^ — ' - -. ~ ~ 



(e - e ) / / /Cf) y) cos cos — ; — dxdy, 



rpab -'o •'o « 



Ao.„ = (e » - 6" <■ )"' / / /('■", y) cos ---^ dady*. 



with a similar expression for ^„|,,, whence the value of d) corresponding to f(,v,y) is known. In a 

 similar manner we may find the values corresponding to the similar functions belonging to each of 

 the other faces. If W be the quantity corresponding to IF for tlie face opposite to the plane xy. 



• The tunclion /(.r, ^) in these integrr.ls may be replaced by /'(.(■, ^ ), since f^ )^ FFcos - - - cos f/j(/^ = 0, unless ;n=H=0, 



