180 
MR. W. M. HICKS OK THE STEADY MOTION AND 
with the aperture, the section will clearly be very approximately circular, and to a first 
approximation the motion will be represented by the stream function found in the 
previous section, the value of x therein being chosen so as to make the coefficient of 
cos v in the expression for the velocity disappear. This will give the first term in the 
expression for the velocity of translation of the vortex, when it moves forward 
without change of form. In order to arrive at closer approximations it will be 
necessary to take account of the form of the section, and this is done in the following 
investigation, so far as to get a second approximation, although the method employed 
is capable of being carried further, of course, with more and more complexity in the 
calculations. 
By impressing on the whole fluid a velocity equal and opposite to that of the 
hollow, the hollow is brought to rest, with the fluid streaming past it. The stream 
function in this case becomes 
xb= 
S 2 
_l r / 2 V 
2 (C-c) 2 1 ttV(C-c) 
2 iy o v /2 ^ \ 
—tw; -zA,,— cos nv 
K n 
where 
A*=2<|(1 
The values of the first three are 
A 0 — {1 + (1 — 
A 
4« 2 — 1 
x- 
4?i 2 — 1 
T' 
A 0 =j[1+^+^+{4(L-2)F+(2L^-^L + 14)^}^ 
A 1 =|[{4-i(8L-15)F-(2L 2 -llL + 17- 1 \)^}^-(l-P 2 -6 1 4^]^“ i !> (19) 
A 3 =^[{16^-4(L-I)^}a;-(P-^)]ri 
The approximation proceeding according to powers of Jc, each coefficient is one order 
higher than the preceding. 
Let U be the velocity at any point of the hollow. Then, to the first order of small 
quantities, where the section is circular 
U= 
”1 b\fr du 
"(C-c) 2 bf~ 
p bio dn 
11=11' 
a 2 S bu 
where 
.rS 2 
2^0 2 (C—c) 2 1 ttV(C-c)P°R' 
v/ ~ ^q^+A^cosv 
The part of U due to the first term is 
Jh_ = 
2^ 0 
x 1—C c 
a 2 C —c 
