gee NATURE 
[AuGusT 3, 1899 
When H is on the tangent at}B or B’, the curve de- 
scribed by points on the axis have cusps; when H lies 
between these points, the curves are looped, and the 
-associated herpolhode has points of inflexion. 
In steady precessional motion «=0, and the focal 
ellipse coalesces with the line SS’; now /=0 and K=4n, 
so that the apsidal angle ae 
ee os 
If » denotes the constant angular velocity of pre- 
<ession, 
(20) 
ee OR. 
OS 
so that 
(21) #_HS” 
m OD 
The steady motion relations 
(22 P=Nu—Axz* cos a, 2=Au sin? a+N cos a, 
when a denotes, instead of 63, the constant inclination of 
the axis to the vertical, will be found, on eliminating p, 
to be equivalent to the geometrical relation 
(23) OQ . OQ’=40D? sin? a, 
so that O lies on a certain hyperbola, xy = {OD?, 
referred to HQ, HQ’ the asymptotes as axes ; thus given 
N and a, a geometrical construction will determine the 
solution of the problem. 
Having selected OD arbitrarily, the hyperbola is 
drawn, and then having laid off HQ’ to scale, draw 
Q’O at right angles to the asymptote HQ’ to meet the 
hyperbola in O ; the tangent to the hyperbola at O will 
meet the asymptotes in Sand S’. 
The second solution will depend on the second point O’ 
in which Q’O cuts the hyperbola. To realise the state of 
motion shown in Fig. 28, the point O on the hyperbola 
must be close to the asymptote. 
It is very desirable that a penultimate case of this 
nature should be worked out completely. We have the 
requisite analysis for the calculation of the algebraical 
case when there are 22 cusps, but the work would require 
the arithmetical courage of a Dewar, now unfortunately 
no longer available. 
Supposing that in consequence of the friction of the 
pivot the motion has steadied down to uniform precession, 
then a eS) tap on the spindle, to (ae) the 
precession makes O move along OQ and produces a 
is ellipse ._ rises 
focal eam) and the axis (ie 
A tap in the vertical plane makes O start out normally 
to the plane, and the lines HQ, HQ’ are now gener- 
ating lines of Darboux’s hyperboloid, in an intermediate 
position. 
Once the limits @, and @; are assigned, and the corre- 
sponding apsidal angle ¥, a regular curve satisfying these 
conditions drawn empirically will give an idea of the 
complete curve described by the axis. 
But if it is desired to plot a number of intermediate 
points from tabular matter of the elliptic functions, we are 
baffled by the mixture of the real and imaginary arguments 
of the theta-functions required in the calculation of W and 
¢, although @ is readily calculated by equation (17). A 
courageous attempt at this computation is made in IV. 8, 
by Herr Blumenthal, but the hidden rocks of error are 
numerous and plentiful enough to make this procedure 
dangerous. Ifthe calculation of a number of guiding 
points between the extreme points of a branch of the 
spherical curve is required, we had better utilise Jacobi’s 
theorem, that the curve described in a horizontal plane 
round the vertical OG by the extremity of the vector OH 
of resultant angular momentum isa Poinsot herpolhode ; 
and having plotted this herpolhode, the vertical plane 
NO. 1553, VOL. 60] 
defined by y may be drawn perpendicular to the tangent 
HK of the herpolhode, while a simple relation of the 
form 
OH@S 2 
eg OD? 2AP 
will give the corresponding value of 6, and hence the 
spherical curve, or its projection, orthographic or stereo- 
graphic, can be plotted to any desired accuracy. 
This theorem of Jacobi is almost self-evident when 
interpreted by means of Darboux’s representation of the 
motion by the deformable articulated hyperboloid. 
In this method the rod OG is held in the vertical posi- 
tion, and the point H is guided round the herpolhode ; 
and ‘then OG, connected by the articulation, will imitate 
the movement of the axis of the top ; for a quadric surface 
can be drawn through H coaxial with the articulated 
hyperboloid, and normal to HP at H, and the squares 
of the'semi-axes of this quadric will be, by a well-known 
theorem of solid geometry 
(25) HQ. HV, HQ. HT, HQ. HP, 
with proper attention to sign; and these being fixed 
lengths, the quadric is a fixed quadric. 
Its equation may be written 
cos 6+2——. =ch 6, + cos @,-++ cos 3, 
(26) Ax? + By? +C2?= D8, 
with 
= DEV DHT Deir 
5=HO; == = = a: 
(27) HO} K7HO’ BTHO’ C Ho '*» 
the + sign being taken according as Q and V, T, P areon 
same . 
the \ aeae ) sides of H. 
An accent will serve to distinguish the case where a 
fixed quadric rolls on the plane through H perpendicular 
to HP’ ; and now we see why the same polhode described 
by H with respect to the axis of the articulated hyper- 
boloid has associated with it two distinct herpolhodes. 
(To be continued.) 
LIFE-HISTORY OF THE PARASITES OF 
MALARIA. 
ie parasites which cause malarial fevers in human 
beings belong to a very homogeneous group, other 
species of which are found in certain bats and birds. 
The life-history of the three species of this group which 
have been completely studied is as follows. The youngest 
parasites exist as amoebule or myxopods within the 
red blood corpuscles of the vertebrate host. Each 
amcebula possesses a nucleus and nucleolus ; and its 
movements vary in extent and rapidity with the species, 
but, in the case of birds, never encroach upon the 
nucleus of the corpuscle. The amocbula increase in 
size ; and, as they do so, tend to lose their movements 
and to accumulate in their ectoplasm certain black 
granules, the pigment or melanin, which are the product 
of assimilation of the haemoglobin of the corpuscle. In 
from one to several days the parasites reach their 
highest development within the vertebrate host, and 
become either (2). sporocytes or (6) gametocytes. 
The sporocytes, which are produced asexually, contain 
spores which vary in number according to the species. 
The spores do not possess any appreciable cell-wall. 
When they are mature the corpuscle which contains 
them bursts and allows them to fall into the serum. They 
then attach themselves to fresh blood corpuscles, and 
continue the propagation of the parasites indefinitely in 
the vertebrate host. The residuum of the sporocyte, 
consisting chiefly of the pigment, is taken up by the 
phagocytes of the host for eventual disposal in the host. 
The gametocytes, while in the blood of the vertebrate 
host, are still contained in the shell of the corpuscle. 
