AucusT 3, 1899] NATURE 321 
A BZ OH?+0S? OQ. HP 
ne Wir SOBELOS: —,00.HP, 
: Wa h = ch @, Ob? > sh 0, Ob? > 
cis ; 20H2— ARB? 
peace V7 P te AD eas =ch 0+ cos 0+ CS 63. 
ite oA, =n 2AP OD: 
: } oe If 0,, 42, 8. iv 
So also C denotes, as usual, the moment of inertia, in 19. ¢72)) ZarGte SS then i ; 
g.cm.*, about the axis of figure, and, with an angular ee. while © % — OHS ROSS 
velocity ~ rad./sec. about the axis of figure, N = Cr, the HS cos @ OH*—-OS” 
angular momentum about the axis ; while 7 is the con- 
stant angular momentum round the vertical Oz. 
In the discussion of the numerical cases, the authors 
start by taking a root « =e of the cubic U = 0 as given, 
and then examining the various relations which subsist 
between N and w. But if one root of the cubic is known, 
the other two are determined by the solution of a 
quadratic, so that we may take all these roots as the 
three data of the question, and follow Darboux’s method, 
as explained in the notes to Despeyrous’ “Cours de 
Mécanique.” 
We may, for symmetry with Weirstrass’ notation, 
denote the two extreme limits of @ by 6, and @,, so that 
cos @, and cos @, are two roots of the cubic U ; the third 
root, being greater than unity, may be denoted by ch 4,, 
and now 
A°*U=2AP(ch 6, —cos 8) (cos 6,—cos 8) (cos @=Ccos 64) 
=2APV 
suppose, with 
ch 6,>cos 6,>cos 6>cos 63. 
It is not clear how any simplicity is gained by con- 
sidering negative as well as positive values of P 
(p. 248) ; this seems to introduce needless complication 
in the classification, as we can take P always positive, 
and measure the angles 6, and 6, from the upright 
vertical position of the top. There may be a slight dis- 
advantage in the case of the spherical pendulum, in 
which the chief part of the motion takes place in the 
lower hemisphere, but the counterbalancing advantages 
of simplicity of classification prevail on the whole. 
In Darboux’s representation of the motion of the axis 
by means of the generating lines of a deformable hyper- 
boloid, we take a focal ellipse, of which the ratio of the 
axes is equal to the modulus « of the real period ; the co- 
modulus x’ of the imaginary period o’ is thus the 
eccentricity of the focal ellipse ; and 
» COS 8,—COS 0, 
1, __ch 6, —cos 6, 
~ ch6,—cos 0’ = Srey E 
(10) kK = y 
ch 6; — cos @3 
The two generating lines HP and HP’ are placed in 
position at an angle @, as tangents to the focal ellipse, 
and the deformable hyperboloid is completed in Henrici’s 
manner by a number of other rods as tangent lines, 
knotted together at the points of crossing. With this 
model we can represent graphically the various constants 
of the problem. 
Returning to the case considered by the authors, 
where @,, N, and 7 are given, we can select an arbitrary 
length OD, and measure off lengths HQ, HQ’ along two 
straight lines inclined at an angle 6., such that 
IE) 2 HO! Naa 
ODE2VAP OD) 2x/AwE , 
and draw the perpendicular QO, Q’O to the lines meeting 
in O. We are now given the conjugate semi-diameters 
OH and OD, by which the ellipse can be described 
through H, and the confocal ellipse, touching HQ and 
HQ’, is the focal ellipse of the deformable hyperboloid. 
On this diagram 
(11) 
OH?-AB? .. OS . OM 
(12) cos 6,= — Opa sin @,=2- Fanee 
cos a Sin 0,= 20 aN 
NO. 1553, VOL. 60] 
from which, when the focal ellipse is drawn, the position 
of H and the tangents HP, HP’ can be drawn. 
The angle AOQ is the amplitude function of a certain 
fraction /K’ of Jacobi’s quarter period K’, with respect 
to the co-modulus x’, the eccentricity of the focal ellipse ; 
denoting AOQ by ao, then, in Legendre’s notation, 
(13) Fow=/K’, 
whence the fraction f can be determined from his tables ; 
so also the fraction 7’ for AOQ’. 
In connection with the dynamical interpretation there 
is an important point L in the tangent HP, such that, in 
Jacobi’s notation for the Zeta-function, 
(14) QL=OAZ/K’ 
LV, LT, LP=OA (zs, zc, zd) fKK’. 
Expressed in Legendre’s notation 
ZfK'=Ew-fE, 
from which the position of L can be calculated by 
Legendre’s Tables ; and now a reduction of the elliptic 
integrals of the third kind in (7) will show that the apsidat 
angle 
HL,- - 
Ssh 4 fm. 
(15) ¥ oak t4/™ 
Integral (6) for the time ¢ gives 
r du Fo OD 
— = F 
me) ss i /2V_ /4(ch 6, —cos 03) OA % 
where 
(17) cos 8=cos 03 cos” @ +cos @ sin », 
a different use of » to that employed in (8). _ : 
Thus if T is the time taken by the axis to swing 
between the extreme inclinations 6, and @,, 
(18) Fae Oe 
OA 
When H is at T, N=, and the rosette curves de- 
scribed by Prof. Klein are obtained, in which the axis 
passes periodically through the highest vertical position, 
shown in Figs. 55, 56, 57. When His at V, N= again, 
but now the axis describes an intermediate path, never 
becoming vertical. When H is at P, N=-x, and 
cos 6,=-—1; the axis now describes a new series of 
rosette curves, all passing through the lowest vertical 
position. These rosette curves are shown in one of Mr. 
T. I. Dewar’s stereoscopic diagrams. 
If we make x=1, the focal ellipse becomes circular, 
and now N=2z=2,/AP cos $6, and the axis reaches the 
highest vertical position asymptotically, the case repre- 
sented in Fig. 58; and the hyperboloid shuts up into the 
axis, like a deformable napkin-ring. ; 
Having settled upon a certain modulus x, and a certain 
fraction f, which gives the tangent TPQ of the focal 
ellipse, we can examine the variety of cases which arise 
by taking different positions of H on this tangent. 
In the spherical pendulum N=0, so that H must be 
placed on the pedal of the ellipse with respect to the 
centre O ; and now 
(19) oa = AES 
3 90H  dn/’K’ 
and similarly ap 
ae. 8, —sn fk’ 
3 enf’K” snf'K’ 
