PROFESSOR W. M. HICKS ON VORTEX MOTION. 
There is no root corresponding to n = 1. The first root is 
X., = 5*76346 = 360° — 29° 46' 41". 
The formula gives for this root 
X = 5*76448. 
For n > 2 it is exact to five places. 
'rhe first three roots are 
5*76346 = 360° — 29° 46' 41" d 
9-09506 = 540° — 18° 53' 29" I.(33). 
12*32296 = 720° - 13° 56' 48" J 
23. Equatorial Axes .—An equatorial axis is the line of particles which remains at 
rest. It is given by the equation 
or by 
f = «. 
when d = 0, 
S - £ =«• 
The positions of the axes are, therefore, readily observed by means of the graphical 
construction in fig. 2, Plate 1. They depend on the abscissm of points for which the 
tangents to the J curve and the parabola are parallel. For values of X > Xi^\ the 
inclination of the parabola to the axis of x is always small. Hence the equatorial 
axes must always be near the crests (or bottoms) of the J curve, i.e., near values 
( 2m + 1) 
The equation for the axes becomes, if y be put for \x, 
1 ^ J' 
cos y + (2/ — - j sin J/ - 2/ -, =0 . 
(34). 
in which the roots < X are required. 
As the values of the secondary cyclic constants and other important properties 
depend on the position of the equatorial axes, it will be necessary to determine their 
values. We shall do this (1) for the case of X small, and (2) for the case of X large- 
As, however, the case of the X, values is special, we shall treat these separately. In 
the case of values other than Xj, say, e,g., Xo parameters, all the axes of any aggregate 
depend on the particular X value. In the case of Xj, however, they are independent 
of the particular Xj. In fact, the successive Xi aggregates may be built up by taking 
any one and putting outside of this a suitable vortex shell. Moreover, the values of 
the axes for the Xi roots are the crests, and bottoms, of the J curve, and so are 
important for their own sakes. 
L 2 
