HYDKODYNAMICAL NOTES 17 method of obtaining it. The results are in harmony; but the fact is not immediately apparent, inasmuch as Stokes expresses the motion by means of the velocity-potential, whereas here we have employed the stream-function. That the subtraction of £«r2 makes (10) an harmonic function shows that the series multiplying r2 can be summed. In fact sin(n7rO/ot) _ cos (2(9 a) 1 vnr (nV2 4a2) ~ 2 cos a 2 ' o 2 so that cos (20 -a) ^rn^asinn7T0/a =-^ - -+8a22 -^ -r1-2 cos a n7r(n~7r24<a2) (11) ^ In considering the character of the motion denned by (11) in the immediate vicinity of the origin we see that if a < |TT, the term in r2 preponderates even when n= 1. When a \TT exactly, the second term in (11) and the first term under £ corresponding to n = 1 become infinite, and the expression demands transformation. We find in this case = ir2 + (0 - ITT) cos 20 + r2 sin 2(9 (-"- log r - ^-\ + 7T \7T 47T/ 7T ......... (12) the summation commencing at n = 3. On the middle line Q = £TT, we have The following are derived from (13) : r -^ r -^ ? -\tf^ o-o ooooo 0-4 14112 0-8 13030 0-1 02267 0-5 16507 0-9 : 07641 0-2 06296 0-6 17306 i-o ooooo 0-3 10521 0-7 16210 The maximum value occurs when r = '592. At the point r = '592, Q = ITT, the fluid is stationary. A similar transformation is required when a = 3?r/2. When a = TT, the boundary becomes a semicircle, and the leading term (n = 1) is = -!:2/---> ..................(14) which of itself represents an irrotational motion. K. VI.id body. The motion expressed that which would ensue if the rotation of the vessel were suddenl; A related problem was solved a long time since by Stokes*, who c the irrotational motion of fluid in a revolving sector. The solution problem is derivable from (10) by mere addition to the latter of i/r0 for then ^ + ^r0 satisfies V2 (^r + ^r0) = 0 ; and this is perhaps th«