Mr. stokes, on SOME CASES OF FLUID MOTION. 129 



1 / 1 d-P\ „ , ^ 



<^=P+Q.--((^.-^).- (.5). 



Now if ^ be the angle between the normal at any point of tlie ellipse, and the major axis, we have 



^ = + 2 e sin 2 0, 



and the velocity of the ellipse resolved along the normal 



= Ccos5= C(l -e) COS0 + C6COS30 (26). 



The velocity of the fluid at the same point resolved along the normal is 



d(h . ^ d(h 



-i- + 2esm20^ , 

 dr rdij 



d<b 2£ . ^ d(b .. 



or -Jf: + _sin2 0--i (27). 



dz c dd 



Let P and Q be expanded in series of cosines of 9 and its multiples, so that 



P = 2°P„cosw0, Q=2°Q„ COSW0, 



there being no sines in the expansions of P and Q, since the motion is symmetrical with respect to 



the major axis ; then 



(P = ^: {P„ + Q„z -^(Q„--P„)^'} cos nO (28); 



g = 2:|Q„ -i(Q„ -^PJ^I cosnO (29) ; 



— ?-?=-2o"4^W^-^)4sin.e (^°)- 



c + z dd (c \ c c- 1 } 



For a point in the ellipse, z = ce cos 2 9, whence from (27), (29) and (30), we find that the 

 normal velocity of the fluid 



p P 



= 2° {Q„cosw0 + -[w(w- 2) -2_ Qjcos(« -2)0 + -[m(w + 2) — - Q„] cos (« + 2) 0} , 



which is the same thing as 



2°|jC«(w-2)^'- Q,,-2] + Q„ + ^[«(» + 2)^'-Q»«]|cosn0 (31), 



if we suppose P and Q. to be zero when affected with a negative suffix. This expression will 

 have to be equated to the value of C cos ^ given by equation (26). 

 For the part of the fluid without the radius y we have 



(p = A^ log »• + 2" — ^ cos n *, 



n 



since there will be no sines in the expression for (p, because the motion is symmetrical with 

 respect to the major axis, and no positive powers of r, because the velocity vanishes at an infi- 

 nite distance. 



From the above value of (p we have, for the points at a distance y from the centre, 



^ = ^-vr!^cos.0, 



dr y y 



" The tirfct term of this cxprcsHion in accurately equal to zero, 

 »ince there is no cxpanHion or conlraction of the solid, (Art, Jt), 

 I have however retained it, in order to render the solution of the 



Vol. VIII. Pakt I. R 



problem in the present article independent of the proposition 

 referred to. 



