1892.] The Potential of an Anchor Ring. 451 



and the 



8c 



charge = 7rA , where a denotes ajc and X = log — — -2. 



5. The velocity potential of a ring moving in an infinite fluid is 

 next found — 



(1) When the ring moves parallel to its axis ; 



(2) When the ring moves perpendicularly to its axis ; 



(3) When the ring rotates about a diameter of its central circle. 



The stream line function is found — 



(1) For motion parallel to the axis ; 



(2) For the cyclic motion through the ring. 



Thus, in the first case it is shown that 



4X + 1 > 2 , 16 X 2 4- 104 X 4-33^1 dl x 12 Xo 4- 9 ^ 4 dl 2 



JT . 4X o + l k 2 . 16 V + 104X0 + 33 , 4 -| dU 12A 



<I> = wo? { 1 H c - H — 



1 L 8 64 J dz 16 dz 



p 4 112\ + 91f ^ 6 ~[ dl z Vril 4 5£ 8 dl s \ 

 Ll6 256 ° J dz 48 dz 1024 dz J ' 



where I, = f cd ±—. , 



Jo \f 0" + c~) ~ 2crsin#cos0 



and I« = [ 2c ^j l % 



and £ = aj'2c. 



The kinetic energy is calculated in each case as far as £ 4 > the 

 result being that • 



2T = P O 2 -f v-) + Rw 2 + A ( Wl 2 4- a- 2 2 ) 4- K/c 2 , 



where 



p _ M f 12X 4-1 72X 2 -24\ 4-7 . 4 \ 



^-yi 1 -^- 6o " ig — *> r 



B « M { l- 4 -Mi V + 16V + 8X -23 tf + a _ y • 



A = M— { i-^±I V +^Z^±l_7 4 ^ J 

 2 I 4 32 J 



^ c f 4X 2 + 8X + l 64 X 3 4- 188 V+ 192Xo-49 , 4 , 1 



^ T I i °" iii 0+ *" J' 



where M is the mass of the fluid displaced. 



