86 
PROFESSOE W. M. HICKS OK VORTEX MOTIOK. 
and 
denote mr by /5 ; then 
X < 77 = ^77 say, 
cc^JX 
= (-)“ P 
3 
. 3 9 
sm ( X + 
3 X \ cos + ^ + ^3 
9 
, ^ , mV ,/ . 2X . 6 + 3X-A ri / ,X\/.^,3 ^ 
+ cos X (|l - - sin X 
2X , 6 + 3X^\ 
3 
X\ 
therefore 
& + 2 = 2 
■-(-)-"ig-|(. + f + HP>.x-(| + f)...x 
1 
For very large values of \ we may neglect powers of —, and then 
P 
pitch = 77^/2 1 — ( — )" 
\:n+n. 
m 
n 
cos X I . 
For the outside shell m = n, 
pitch = 277 sin |X. 
Thus in the case of the Xo aggregates the pitch of the outer layer is very small. 
If we number the shells backward from the outside, we write — ^9 + 1 for m, 
and the pitch is 
77^2 { 1 + ( - )- (''* ~p - + - y cos xV. 
It is seen, therefore, that tliere are two series of shells in aggregates of large X, one 
in which the pitches increase as we pass inwards, and an alternate series in which it 
decreases. If X lies between a X^ and a X.o parameter (Xo > Xi), the outer series belongs 
to the first category. If X lies between a X., and a Xj value (X^ > Xo), the opposite is 
the case. In other words, if the parametral point P in fig. 2, Plate 1, lie above the 
line of abscissse, the outside layer has a very small pitch, and those of alternate_shells 
increase as we go to the centre. If P lie below the opposite is the case. 
The vortex spirals are given by 
<?dx 
{(J - P-J') (J - PJ' - 1>)Y 
