176 
MR. W. M. HICKS OK THE STEADY MOTION AND 
05= — {1 + (1 — X)a5}~° + 2^( - )*{ (4 n z -1 +\)x— l}^r 
■IX o JX n 
aJ |l + (l_X)|i+2S 1 (-)*- 1 (4#-l+X)^} = -|f + 22 a (-)-ip > 
The right hand member of this equation is the velocity at the centre due to the 
cyclic motion alone, divided by 2xjj 0 /a 2 . Call this velocity V l5 and denote the critical 
value of by V 0 , then 
l'= 1 + (1 -X)^?+ ■2S 1 ( 1 +A)b* 
V 0 JX 0 JX« 
The most important terms in these expressions are 
x »=2(^{ 1+ ( SL_6 - te_ h^) / ‘' :+ 
^' = l + 4^F+ . . . 
v o 
The stream lines will be given by 
\fj—±Yp~= const 
and by choosing the constant properly, we may make this represent the surface of the 
fluid carried forward. To determine the constant we need only find one point on the 
surface by the above method. If the value of x is less than the critical value, the 
siu’face will extend to the axis; in this case the best way will be to put u== 0 and 
find v from the equation 
V= 
1 b\fr du~ 
_p du dn\ u={) 
If on the contrary x is greater than x 0 , the surface is ring-shaped, and it will be 
best to find u from the equation 
”1 b-yfr du 
p du dn 
V= 
V—i r 
If x be negative, or the velocity of translation and the cyclic motion wit hin the 
aperture be in the opposite direction, the corresponding equation will be 
difr du 
du dn 
_ y=o 
In tabulating corresponding values of u, v and V/V 0 the best w r ay would be to 
insert values of u, v and determine V/V 0 . The following numbers in the case of 
h— sin 1° were obtained in this way. For the case of x less than the critical value, 
the surface cuts the straight axis at points given in the several cases by v, 
