Mr. stokes, on SOME CASES OF FLUID MOTION. 133 



whence 



putting 2„, for shortness, to denote the sum corresponding to odd integral values of ?i from 1 to « . 

 It is evident that the value of <p corresponding to the motion of the opposite face in con- 

 sequence of the angular velocity w" will be found from that just given by putting b - y for i/, 

 and changing the sign of w'"; whence the value corresponding to the motion of these two faces 

 in consequence of u>"' will be 



itnit nny nnh n^y 



ia)"'a^ 1 (£~ — l)e »~ + (e"~ — l) e " Wtt.* 

 ; — ^0 ~i J. ; cos . 



e o _ £ a 



Let this expression be denoted by w"'\p(v, a, j/, b). It is evident that the part of d> due 

 to the motion of the two faces parallel to the plane yz will be got by interchanging .?■ and i/, 

 a and ft, and changing the sign of to"' in the last expression, and will therefore he-co"'\jy(y, b, x, a). 

 The parts of cp corresponding to the angular velocities w, w", will be got by interchanging tlie 

 requisite quantities. Also the part of (p due to the velocities U, V, W, will be Ux + Vy + Wz, 

 (Art. 7), and therefore we have for the complete value of <p 



Ux + Vy + Wz + w" {^lr{x, a, y, b) - v//(_v, 6, x, a)\ + tu'{\//(2/, 6, «, c) - >|/(ar, c, y, b)\ 



+ w"\\Ij{z, c, X, a) - \//(,r, a, z,c)\. 



According to Art. 7 we may consider separately the motion of translation of the box and 

 fluid, and the motion of rotation about the centre of gravity of the latter ; and the whole pressure 

 will be compounded of the pressures due to eacli. The pressures at the several points of the box 

 due to the motion of translation will have a single resultant, which will be the same as if the 

 mass of the fluid were collected at its centre of gi-avity. Those due to the motion of rotation 

 will have a single resultant couple, to calculate which we have 



(p = u)"'{\/.(*', a, y, b) - xf^iy, b, x, o)| + &c. 



Since for the motion of rotation there is no resultant fm'ce, we may find the resultant couple 

 of the pressures round any origin, that for instance which has been chosen. If now we suppose 



the motion very small, so as to neglect the square of the velocity, we may find —J- as if the 



axes were fixed in space. We have then for the motion of rotation 



dw , 

 p = - P—^ l^C''. a, y, b) - \jy{y, b, x, a)\ - &c. 



11 11 1 /. 1 • . 'f'" ^^"> '^<" 



Hence we may calculate separately the couples due to each of the quantities — — , ,- and — — . 



It is evident from the symmetry of the motion that that due to will act round the axis 



of z, and that the pressures on the two faces perpendicular to that axis will have resultants 

 which are equal and opposite. Also, since >// (a, o, y, b) = - xj/ (0, a, y, b) and >J/ (x, a, h, b) 

 = - \j/ (x, a, 0, b), it will be seen that the couples due to the pressures on the faces perpen- 

 dicular to the axes of x and y will be twice as great respectively as those due to the prcssiu'cs 

 on the planes yz and xz. The pressure on the clement dydz of the plane yz will be p^^„dydz, 

 and the moment of tliis pressure round the axis of z, reckoned positive when it tends to turn 

 the box from x to y, will be 



