PROFESSOR W. M. HICKS OK VORTEX MOTION. 
77 
The roots beyond this are therefore given by the formula cori-ect to live places. 
The radii of the equatorial axes are r = yajX. Hence using- the values of given 
in (31), the tirst three are. For X(^\ 
2-74;37l 
r — 
4-49341 
a = -61062 a. 
For Xf, 
2-74371 , 
■i\ = ^3-^3 « = 35516 a J 
>-• 
h or Xf 
9- 31663 
10- 90408 
6-11676 
1F90408 
_ 2-74371 
~ 10-90408 
a = -85442 a 
a = -56096 a )> • 
a — -25162 a 
Case of X large. —The number of equatorial axes depends on the order of the X2 
parameter next greater than X. If X lie between and \f\ there are n such axes. 
It seems then natural to refer the magnitude of X to Suppose then 
X = Xi“> - X, 
where the maximum value of X is about tt,— or we may write X = + X, and if 
both be allowed X will have a maximum of the order 
— nrr 
The equation in y is 
cos y -\-[y 
in which the first n roots are to be determined. For small roots the parabola of fig. 3 
is almost coincident with the axis of x, and consequently the small y roots are very 
nearly equal to the corresponding values for X^. It will be best to obtain an 
expression for the large roots and then see how far back it holds for the smaller roots. 
Clearly y is always near m-rr where m is an integer < 71 . 
Put 
y = rmr z = a z say. 
Then 
C0S2: , 1\ . / ^ 'I\ 
J (X) may be either + or —, it is of order of magnitude 1 at most. 
- A_ i/AY_ i/sy. 
wtt ^ ymr/ ^ \ TT / 
- -jsmy-2/^ = 0, 
