68 
PROFESSOR W. M. HICKS OH VORTEX MOTIOX. 
But 
Hence 
T n 
- y ^ = 0 - 
Tvr — ^__ 
m- 
3 N y'n y'n—l 1 (^) O ■ I ■ / O O \ 
Va-i) - - - - y-’ y-- + yy^ sm y,,_, . (23). 
20 . The velocity of translation is given by 
U= - 
/A \ sin X — 3 (sin X/X — cos X) 
3« X (Six — sin X) 
fjL cl 
3a (Six — sin X) d (kcc) 
{J (Xx) - x-J 
To see how this varies with the parameter X, refer to the graphical construction in 
fig. 2, Plate 1. The curve J and the parabola intersect in P. If A be a point on 
Fig. 3. 
the curve (fig. 3), and B on the parabola with the same abscissa near P, and PN be 
the perpendicular on AB, 
Tj _ _ d _^ 
3a (Six — sin X) PH 
_ fjL sin (« — ;d) 
3«(SiX —sinX) cos a cos jd 
fx sin (a — /S) 
3aA cos « cos d 
Where a, /3 are the angles which the tangents to the curve and the parabola at P 
make with the axis of x, and A denotes the area of the curve OAPMO. 
The factor o always finite, except for X = 0, and positive. It is 
then easy to see in general how the velocity alters as the parameter X increases. 
As P (fig. 2, Plate 1), travels along the curve, U is positive. Leaving out of sight 
for the present its value for X small, it later on diminishes to zero when P reaches a 
