90 
PROFESSOR W. M. HICKS OH VORTEX MOTIOH. 
The first integral is 
O'} do: 
J - o}3' 
Now J - x-y 
does not vanish. 
= x‘ {\ — .x')F [x), where F {x) is finite for x between 0 and 1 and 
Hence 
_ dx 
0 (1 — x) (1 + oj)Fx 
, = X + XJ' I 
J I 
= X+XJ'f^ 
= X + 
Therefore 
dfjdx 
log + finite quantity, 
" = + lik ’°S 7 + 
X sin X — 3 J ^ s \ -t J — orj 
,,.2T' "1” 
d. 
\ dx, 
where s is the distance from the pole of the point at which the stream sheet \p cuts 
a line joining the pole to a point on the equator. The angular pitch is therefore 
infinite at the surface owing to the filaments being parallel to the equator at points 
close to the pole. 
25. Graijhical Methods, —The graphical construction indicated in § 17 aftbrds a 
very convenient method of obtaining a general qualitative view of the properties of 
these aggregates. It serves also for a rough quantitative one, and at least gives for 
many determinations the rough starting jDoint which is always the most troublesome 
obstacle in numerical approximations. It may be well, therefore, here, to collect 
and enlarge on what has gone before in this respect. 
The first thing is to trace on a large scale the curve y = J (X) where X is the 
abscissa. This is very easily done, since J is expressed in simple functions which are 
tabulated. The curve is drawn for the first three undulations in fig. (2), Plate (l). 
Now X determines completely the nature of the aggregate (except its volume and its 
intensity). The point P on the J curve, corresponding to X, we will call the para¬ 
metral point. Draw through P a parabola touching the axis at the origin. For all 
points beyond the first few undulations a circle will suffice, or the curve drawn by a 
thin lath bent to touch the axis at 0 and to pass through P. If x denote r/a, Xx 
will correspond to a point on the J curve between 0 and P. If Pj, Po denote the 
corresponding points on the J curve and the parabola, the value of xfj in the aggregate 
at the point (r = xa, 0) is given by P1P2 siird (note P2P1 will be negative). The 
velocity of jjropagation will depend on the angle at wdiich the parabola and curve 
intersect at P (see fig. 3, § 20). If they touch, the angle is zero, and the translation 
velocity zero. In fact the parameters of the points are the Xo values. We will call 
