132 Mn. STOKES, ON SOME CASES OF FLUID MOTION. 



(p due to any one of the angular velocities, we may calculate separately the part due to the motion 

 of each face. 



Let the origin be in a corner of the box, the axes coinciding with its edges, Let a, b, c, 

 be these edges, U, V, W, the velocities, parallel to the axes, of the centre of gravity of the interior 

 of the box, ft)', ft)", ft)'", the angular velocities of the box about axes through this point parallel 

 to those of w, y, x. Let us first consider the part of cp due to the motion of the face ,ra: in 

 consequence of the angular velocity w '. 



The value of (h corresponding to this motion must satisfy the equation 



-rr + -ri^ = {36), 



with the conditions 



d(p 



0, when X = or o (37), 



d.v 



-i = 0, when « = 6 (38), 



dy 



-^ = 0,'" (.V--] , when 2/ = (39), 



dy \ 2/ 



within limits corresponding to those of the box. 



Now, for a given value of y, the value of d) between x = and x = a can be expanded in a 



convergent series of cosines of — '- and its multiples; and, since (37) is satisfied, ihe series by 



which -J- will be expressed will also hold good for the limiting values of ,r, and will he conver- 

 dx 



gent. The general value of (^ then will be of the form 2" r„ cos . Substituting in (36), 



and equating coefficients of corresponding cosines, which may be done, since any function of .r can 

 be expanded in but one such series of cosines between the limits and a, we find that the 



general value of Y„ is Ce « +C'e " , or, changing the constants, 



mtt(/i- v) i}ir{h-y) rnrjf nwij 



r„ = J„ (e '■ +e' ■' ) + B,, {e - + e' ■■ ), 



when 11 > 0, and for n = 0, 



Yo = J„y + So- 

 From the condition (R8) we have 



TT ^ — J— }l7rX 



Ao+ -'2i nB„ (e •' -e " ) cos = : 



a a 



whence J„ = 0, B„ = 0, and, omitting B^, 



nwjb-y) THr(ft-;/) y^ ^ ^j, 



(b = 'Ef A„(e » + e' " ) cos . 



' a 



From the condition (39), we have 



2i nA„ (e " — e " ) cos = ft) \x . 



a ^ 'a \ 21 



Determining the coefficients in the usual manner, we have 



