PROFESSOR W. M. HICKS OH THE THEORY OF VORTEX RINGS. 
733 
If there are no added rotations A 0 , A ' 0 will therefore be of order Td with reference to 
A a 4 . By always taking them to be of this order therefore the approximation will 
still be true when there are added circulations. 
4. It will be convenient here to consider the equation giving the pressure at any 
part of the core, including the case where small vibrations are superposed on the 
steady motion. The equations of motion are, d being the density, and taking the 
motion in the plane of z, x, 
1 8p 8u 8u 8u 
-dte = st +u hi +w '& 
1 8p 
d8z 
8w 
8t 
8w . 8w 
ox ox 
8u 8 vj _^ ^ 
Therefore, integrating along a line in the plane of x, z, 
dx+ i dz)=l(udx+wdz)+u ( S £ dx+f z dz 
-f- w (~ dx +y dz ) + A x(udz — wdx ) 
or if v denote the velocity at any point and v the velocity along the line of 
integrations, 
1 8p 8v' 1 8(v z ) . 8\]s 
~dYs = Si~^^~Ss A ~8i 
Therefore 
.(n) 
where <f> is the flow along any line in a plane through the axis of z up to the point in 
question. It is a function depending in general on the path of integration, but <f> is 
independent of this (is in fact due to the added irrotational motion) and is single 
valued when xjj is so. When xp is many-valued (as, for instance, in the case of 
pulsations), <j>—Axjj is single-valued. 
Section II.— Steady motion . 
5. Suppose the approximations are carried so far as to include terms in cos nv in 
the stream functions and equations to the bounding surfaces. The constants to 
be determined will then be V, the n— 1 quantities a, the n quantities /3, and the 
coefficients A'„, A n , B„, 3 (w-f-l) in number—or in all 5n-j-3 quantities. We may 
MDCCCLXXXV. 5 B 
