392 
PROFESSOR M. J. M. HILL ON THE MOTION OF FLUID, PART 
Calculating 
47 r\clx~ dif) Iir'dx'-' df~ 
_ i n_i\f-2f t 2/ 
47t\5 2 a 2 J\a 2 Ir 
o - ! - . 1 —2d) 
*y 
\ cr o~ / 
__£_/l_ I 
*y 
4tt\5 2 a?J(^ y 2 Y~ 
* 2 + J 2 
The value of the potential due to this density is the single-valued expression given 
in Art. 15, viz. 
x' 
c a —b , , cab . . , 
S'VPi^^+Tl 81 ' 1 
x' 
' 2 Sm x/x'1 + y' 2 
Add to this the term ~ sin 1 ^ to give the cyclic constant —■ (— 27r) = 7ra&.2£ 
\A' 2 + ?/ 
in Kirchhoff’s notation ; then, in order that what is now obtained may be the same 
as y, it is necessary that 
„a 2 — b“ _ c a — b 
•' a % 2 = 2 a + b 
7 / . / A Crt5 
- a T~a-f = T 
the latter of which is known to be true, but the former will not also be true unless 
W ah' 
cab 
o=2£- 
ab 
in Kirchhoff’s notation. 
(a + b)~ (a -f bf 
Up to this point no relation has been assumed between cb and f 
Supposing, however, this relation satisfied, the velocity potential at an external 
c a l $£ , 
point is obtained by adding the same term — sin -1 to the potential found i 
Art. 15 for an external point. 
Thus velocity potential at an external point is 
m 
cabx'y' 
a 2 +5 2 + 2e 
-b°~ YV(« 2 + e)(7) 2 + e) 
. . cab . , 
1 +TT sm 
\/ « 2 + e 
• CC * 'l l * • • 
where e is a root of the equation - ■ -f- ~; —= 1, the axes x, y' turning round with 
angular velocity d>= 
2 Sab 
(a + b)~ 
The expressions for the velocity, which may be deduced from this, might also have 
been obtained by Helmholtz’s Method from the current function A which is the 
