70 
PROFESSOR W. M. HICKS OX VORTEX MOTION. 
Hence 
also 
Equation (25) shows that when J' = 0, i.e., for the X, parameters, the stream lines 
and vortex lines coincide. (It is to be remembered that we have supposed in the 
foregoing’ that (/> and x. opposite sides of meridian lines, and therefore 
tan (f) = — tan x means that they lie on the same side and coincide.) 
I. From X = 0 up to X = Xj^^ J > x^J' and J — x%J' < J. Hence between these 
limits, the stream lines and vortex line are on the same side of the meridians, and 
y>^, i.e., the stream lines lie between the vortex lines and meridians. At X = Xi^^ 
they coincide. 
II. Between Xi^^ and Xl^^ J > but J —x^J'>J. For any given X, J changes 
from + to — as x passes through the value \x = For this value of x, or 
r = ^ a, X— b- Thus, for an aggregate whose parameter X lies between the first Xj 
and Xo roots, the vortex lines lie between the stream lines and the meridians for all 
points at a less distance from the centre than r = ^ a. At this distance y = 0, or 
the vortex lines coincide with the meridian planes, and beyond this distance up to 
the boundary the vortex lines and stream lines are on opposite sides of the meridians. 
For values of X between tlie first and second X^ parameters we have to deal with 
two layers. In the outer J — x'J' is negative, whilst J is negative between Xf^ and \f\ 
positive between and Referring to fig. 2, Plate 1, let the point where the 
parabola cuts the J curve be given by X', corresponding in the aggregate to a distance 
from the centre Xx = X' or r = — a. It is clear that J (X) and J (X') are of the same 
sign. Hence, if X lies between X 2 ’ and Xf^ (corresponding to P between Qi and R.,), 
lies between X^/^ and X 2 ^ whereas if X lies between and X' lies between 0 and 
Xy^—or, taking closer limits still, between it and XjE We find, therefore, the follow¬ 
ing results. 
III. P between Qj and R,. In the inner spherical nucleus the vortex lines lie on 
the same side of the stream lines as the meridians—they are, in fact, exactly similar 
to the second category. At the boundary between the central nucleus and the outer 
layer (/> = 0, the stream lines coincide with the meridians, In the outer layer the 
stream lines lie on the other side of the meridian, with the vortex lines beyond. When 
P coincides with R 2 or X is the second Xj parameter the stream lines coincide with the 
vortex lines again, but on the opposite side of the meridians. 
IV. For P between R 2 and Qo, we get still two layers, the boundary being given 
, , X J'x- ,sin 6 
tan y = - tan - — qyy- 
« — J (J _ giiP 0 
(hi [ 
. ( 25 ), 
tan (j) 1 , o 't' 
, ■ — — 1 X . 
tan y J 
