AucusT 10, 1899] 
analytical theorems of the curve. 
OF are the conjugate diameters parallel to the tangent 
HK and the vector GH, and if OK is the perpendicular 
on HK, then, with Darboux’s notation, 
a+6+c=P, bc+ca+ab=Q, abc=R, 
and putting 
OE2+OF2+OH?_ P 
TOD: a? 
OE? . OF* sin? EOF + OK? . OE?+ OF? OG?_ 
- a OD+ i 
OG?~ OE”. OF* sin? EOF_ R 
(3) OD® Recs 
The elimination of OE”, OF?, and sin? EOF between 
these equations gives the relation connecting OH? and 
OK? in the herpolhode ; it is linear in 
OK* and quadratic in OH*. The 
geodetic radius of curvature of the 
polhode on the polhode cone is 
readily found by a differentiation by 
exactly the same formula as the 
vrdr/dp formula of a plane curve; 
and the radius of curvature of the 
herpolhode in the plane of G is the 
projection of this geodetic radius of 
curvature. 
At a point of inflexion on the 
(29) 
a 
m? 
(30) 
Thus, if OE and 
MATURE 
347 
But as these numbers bring the point P inconyeniently 
near to A, fresh dimensions are chosen ; we can take a 
scale such that OA =10 cm., OB=5, so that x is 
reduced from o'521 to o’5 ; and the points P and P’ were 
so placed as to make 6, = 45°, 6. = 30°; and now, by 
| Measurement, OD = 17°73 cm., 
HQ=16'4 cm. HQ’= 8'9 cm. 
ii — eS 4acem: HT’=21'6cm. 
HV=15'0 cm. HV'= 8:3 cm: 
Ite = 36-2/ems HP’=10'4 cm. 
The angle AOQ or » = 33° by measurement, so that 
from Legendre’s Table IX., to the co-modular angle 60°, 
Fw=0'6009, K’=2'1565, 
Bo=0'5528, BH’ =1:2111; 
and thence 
f= — 02787, 
and 
QL=2"153 cm. 
herpolhode this geodetic radius of 
curvature is infinite, and now the pol- 
hode is a bit of a geodesic on the 
polhode cone; this shows that the 
osculating plane of the polhode is now 
perpendicular to OK. 
But to find whether the herpolhode 
can have points of inflexion, we merely 
require to find where the value of OK 
is stationary, and this is found by 
solving the quadratic in OH’, and 
examining its discriminant; in this 
way we shall find that the discrim- 
inant vanishes, and OK is at a turning 
point, when 
on GK? 
a OD? 
where, in Darboux’s notation, 
02=(0?- 4R(P-2). 
The value GR=0 is excluded when the rolling surface 
is an ellipsoid; it will be found that the other value 
makes 
4(a—h) (6—’) (c= 2)R 
=) 1) 2 
i oh 
Mm 
? 
OF2_Q 
m—— i 
OD? 2h 
-OF2 OG? __, 
ye 5 — Lee 
OD+ =~ 
and the maximum value of this being a4, it follows that 
3G eee = t,o HQ" Dae aK 
ge ae aR a 5) 2” OD? ABC Ie =A= B) 
is positive, so that the rolling quadric cannot be the 
momental ellipsoid of real positive matter for points of 
inflexion to exist, in accordance with the theorems of 
Hess and de Sparre. 
Fig. 1 has been drawn with the idea at first of giving 
the graphical representation of the numerical case dis- 
cussed in VI. §6; so that 
fae —0°0068, fas 1°421 
@ w 
(these numbers appear to show that s and s’ on p. 481 
must be interchanged). 
NO. 1554, VOL. 60] 
) 
Thence the apsidal angle, as a fraction of a right 
angle, is given by 
Ww _ AL I 
4x OA 
As 6 diminishes from 6, to 6, the deformable hyper- 
boloid opens out from the plane of the focal ellipse 
and flattens again in the plane of the focal hyperbola. 
As a typical intermediate position choose that of half- 
time ; with our dimensions this will make 
(33 +f=1'245. 
Ba bee 2° 
sin Y TT eo 5359, 
cos 6=0'792, 0=37° 37’. 
Project the figure on to the plane of the generating, 
lines HQ, HQ’, which are now drawn inclined at 37° 37’, 
and place the points Q, T, V, P, &c., on these lines 
as before ; the confocal quadrics project into confocal 
conics. 
The lines TT’, VV’, PP’ form a triangle XYZ, the 
sides of which are the traces of the principal planes : 
Q, the orthocentre of this triangle, is the projection of 
the origin O. 
We must be content with the mere statement of the 
following geometrical theorems, required for throwing 
light upon the analytical theorems of this dynamical 
| problem, with a view of obtaining a clear image of the 
motion in accordance with Poinsot’s ideas ; the demon-. 
strations will be found in Salmon’s “Solid Geometry,” 
