348 
NATURE 
| AucusT 10, 1899 
or better still, in Mannheim’s ‘‘ Géométrie Cinématique,” 
which has been found valuable in carrying on the Poinsot 
traditions of dynamical presentation. 
Join H@, cutting the sides of the triangle XYZ in 
a, B, y ; describe the circle round XYZ, and let XQ, YQ, 
ZQ cut this circle in d, e, f; then ad, Be, yf intersect the 
circle in one point F. F is the focus, and HQ the 
directrix of the parabola, which is the envelop of the 
normals of the confocals (Salmon, “ Solid Geometry,” 
§ 177); this parabola touches the sides of the triangle 
XYZ ; the tangents to the parabola from H are tangents 
to the projections on the plane of the confocals through 
H, while the tangents to the parabola from © are the 
axes of these plane confocals ; also H® and HF are 
equally inclined to HQ and HQ’. 
If He, He are the tangents to the parabola from H, 
the points of contact c and ¢ are the centres of curvature 
of the plane confocals through H; while C and C’, the 
centres of curvature of the normal sections of the con- 
focal surfaces through H made by planes through the 
line of curvature traced out H, are such that 
HC=He cos? 40, HC’=He' sin? $8, 
and thus the point C and C’ are easily constructed 
geometrically. 
The line CC (a tangent to the parabola) is Mannheim’s 
axis of curvature, and the plane through H perpendicular 
to it is the osculating plane of the polhode. Thus in the 
degenerate case of Fig. 1, when H lies in the plane of 
the focal ellipse, PP’ is the axis of curvature of the 
polhode, while Z and & coincide with O. 
The various relations in the conjugate Poinsot move- 
ments, investigated by Darboux and Routh, and given in 
VI. § 8, can now receive a geometrical interpretation. 
Thus the relation (2), p. 476, written in the form 
4 BiB MeCeiCawa 
(34) ACA Are &e., 
is the equivalent of 
Wir ELV ei Vires ELVEN are 
(35) , ke. 5 
Pi ei Ps Ey 
expressing the fact that, if PP’ cuts OA in X in Fig. 1, 
then MX is the harmonic mean of HT, HT’ and of Hm, 
Hw’, where 7, m’ are the feet of the ordinates from P, 
P’ on OA. 
Again, if X@ meets YZ in D, then OD and OV are 
conjugate diameters in the principal plane OYZ of the 
rolling quadric 
6 en | eG 
(36) HVT * HP 
and therefore 
HP'_ fan VOD tan YOV = Y*; 
T VZ 
similarly 
BOY WA eR Ie 
Hib guExX=. HV» PY” 
so that, drawing Xx parallel to YZ to meet HQ and 
HQ’ in x and 2’, 
TV) SIPVaRPVie «HV! PV; 
EDP Semmbes. VT ape 
be VIP AVSIY eaeeenrn ise! VDA 
(37) Hv? ~HV7HW  £4#V®@ 
the geometrical equivalent of the relation 
on (1A) Ma(s=S)(+- 8} 
Similarly for the other relations, which we have not 
space to develop. 
The A and C employed here require to be carefully 
distinguished from the values referring to the top itself, 
which ought to be differentiated by a suffix. 
The constancy of the perpendicular from the centre 
NO. 1554, VOL. 60] 
on the tangent plane of the rolling quadric along the 
polhode is expressed by 
x yp 22 
<a = 
(39) HV?’ aT?’ He?” 
and 
ee eI  % 
Hv’ HT’ HP 
are obviously the cosines of the angles which the line 
HQ makes with the coordinate axes. 
The Darboux-Keenig’s arrangements by which the 
polhode, herpolhode, and associated top motion are pro- 
duced mechanically by articulated movements, are worth 
mention and study in elucidating the various theorems. 
When the parameters a and ¢@ (p. 263) are aliquot 
parts of the period o’, the multiplicative elliptic functions 
a, 8, y, 5 become algebraical functions of , qualified by 
exponential functions of the time. Take the simplest 
case, where a=}, 6 =e’, equivalent to placing P 
at A and P’ at B in Fig. 1, then we shall have 
—_— — = . 
C— eee ——" 
a[e 
I 4 n : by 
a= Tot) Ie" e —u) +2" ** Mua) |, 
V2 Nis 
- : 
B= aul Je’ -u.e'-—u)—2 Ei viu—e) | » Ke, 
/ A/K 
. v2 
with 
peels (2-3) 
ae CLA 
poiek -(Z-N. 
anon Ac a) 
Aésin be¥' =aB=eltV ey /(1 — ze) +2 J(u. k—u)). 
In the next case, where 
a= sw’, b= hw’, 
it is found that we can put 
é=— I +¢ é= — ie +2 a TiS) -! 
? (1—c)? b} Cc LA 
2 ey ,)\2 4 
ne | ee c) +iyv| 
2 (1-c)AJ/e 
eel rel 2022 Nel? 5 
=e es eu. ue 
B 2 GaaNe v( an 
+2{u— ait =)" N(e’= «i &e. 
c 
with 
2 
j=4 2s m+3(2-3)N 
“(T—c)n/2e TNC 
v=- 1(2=<¢) (1=2¢) -4 (Z = <u) 
+ (@=e)W2c NG eA 
The general form of this solution can now be inferred. 
but it is evident that the algebraical complexity mounts 
up very rapidly ; with 
2re' ro’ 
a= —; b =- 
n n 
nt 
eee eel Ay 4tihy JV fs 
2 
> 
1 
B= "(Bi J (e"—u . u—e) +2Bo/(e’ - oy 
V2 
where the A’s and B’s are rational polynominals of z. 
Thus for 7=5, we can take 
of 2c peGassaes es 1 Pym: 
7 Wao re “2 7 Wt. 
NI GEria 4c JC=1 
where 
C= aera; 
and 
Ae Daa 
A, = Pre SP pee eos ESS 
1S el mala ocr T)(c— ae 
