320 
NATURE 
[AucustT 3, 1899 
Trochus, cujus partes cohaerendo perpetuo retrahunt sese 
a motibus rectilineis, non cessat rotari, nisi quatenus ab 
aere retardatur.” We can translate /vochus as a top, 
as well as a wheel or hoop. 
But Newton lived too early to calculate any quantitative 
explanation of the theory of the top, and even his attempt 
at the simpler problem of precession was not a very great 
success. (Principia, lib. III., Prop. xxxix.) We had to 
wait for D’Alembert to straighten out the difficulties of 
first principles before arriving at an exact solution. 
The key-note of the present treatise is found in § 2— 
“Analytische Darstellung der Drehungen um einen 
festen Punkt ”—and in the bilinear relation in equation 
(5) on p. 25, % 
(1) eer with a3 — By=1, 
expressing analytically the displacement of a rigid body 
about a fixed point. 
If x, y, z are the coordinates of a point with respect to 
axes fixed in space, and X, Y, Z of the same point with 
respect to axes fixed in the body, 
eee rs 
—xt+yi 
while A is the same function of X, Y, Z; thus a and A 
play the part of a stereographic representation of the 
point on the sphere of radius 7 with respect to the poles 
in which the sphere is intersected by the axes Oz and 
OZ. 
Expressed by Euler’s unsymmetrical angles 6, o, , 
(2) r 
r—-=z 
} : BUSTERS 
(3) a =cos Mau B=i sin 16c A(p— We 
y=? sin yoch me 3= cos em Sees 
satisfying 
a5=cos” 46, By= -sin? 46, 
and 
ad — By=1. 
By putting 
(4) a=D+Cz, B= Bese 
y=B+ Az, s= D-Cz, 
so that 
A?+ B?+4 C?+D?=1, 
the versor-quaternion 
(5) Q=Ai+Bj+Ck+D 
is obtained, which determines the displacement; this 
gives the authors an opportunity for an excursion into 
the Quaternion-Theory so far as required in their subject, 
and should delight the soul of Prof. Tait, always anxious 
to see more frequent applications of his favourite subject, 
to which allusion is made on p. 509. 
In Maxwell’s opinion, it is the introduction of the 
ideas, as distinguished from the operations and methods 
of Quaternions, which is valuable; and now we have 
McAulay’s “ Octonions” to assist, with plentiful illustra- 
tions of the dynamics of our subject. 
The important use of the above bilinear relation be- 
tween A and A, for the treatment of the motion of a 
rotating body or a top, was pointed out by Prof. Klein in 
a lecture at Gottingen University in the winter-semester 
of 1895 ; the further development of the formulas was 
chosen by Prof. Klein as the subject of his Princeton 
Lectures in October 1896; and the present work is in- 
tended to be a complete presentation, with the collabor- 
ation of Prof. Sommerfeld. 
Subsequent historical research, described on p. 511, 
has shown that the germs of similar ideas can be traced 
back through Hess (p. 429), Weierstrass (p. 511), up to 
Gauss in 1819. Gauss appears at the same time to have 
had some prophetic inspirations of the Quaternionic 
Theory, but as usual he carefully bottled up his ideas. 
It was said of the works of Friar Roger Bacon in the 
middle ages—“ partim mutili direptis hinc inde quaterni- 
NO. 1553, VOL. 60] 
onibus facti, tam raro comparent, ut facilius sit Sibyllae 
folia colligere quam nomina librorum quos scripsit ”—and 
the same might be said of Gauss’s unpublished manu- 
scripts, now at length to be edited completely by the 
Gottingen Academy of Sciences, under the direction of 
Prof. Klein, the present occupant of Gauss’s chair. 
We think then that Prof. Klein is tao generous in re- 
nouncing the priority of his discovery, considering that 
he was the first to make the invention really work, and 
that his precursors allowed the germs of the idea to 
pass from them unfertilised, not perceiving their real 
importance. 
In the special case of the symmetrical top the func- 
tions a, 8, y, 6 can be expressed by elliptic-theta func- 
tions; Prof. Klein calls them ‘“ multiplicative elliptic 
functions” ; their form is given explicitly on p. 520, and 
now the solution is complete from the point of view of 
the mere mathematician of the school of—‘ Shut your 
eyes and write down equations.” 
““Mais il faut convenir que, dans toutes ces solutions 
on ne voit guére que des calculs, sans aucune image 
nette de la rotation du corps.” 
The authors, giving heed to this warning of Poinsot, 
devote the rest of the book to a careful examination 
and classification of the various cases which may occur ; 
and also to what the ordinary mathematician in general 
so cordially detests, the working out and drawing in a 
diagram of some well-chosen numerical cases ; only in 
this way is it possible to make sure of the accuracy and 
reality of the abstract formulas, and to lift mathematical 
science out of the arid collection of analytical results. 
A first great requirement for the study of the move- 
ment of the top is an actual model, as shown on p. 1, 
or else a top such as that devised by Maxwell (‘* Collected 
Works,” I. p. 248); sufficient rotation can be imparted 
by twirling the spindle by the finger ; and a slight blow 
with the hand will give any desired variety to the pattern 
of the curve described by the end of the axle. A bicycle 
wheel, spinning in its ball bearings, supported ina fixed 
cup, would also serve the purpose. i 
As we have three independent constants at our dis- 
posal with the top, the number of possible cases is three- 
fold infinite (co*) ; and so the choice of a numerical case 
is at first sight an embarrassing one in its variety. 
In the book the constants employed are such that 
(p. 223) 
(6) t= [Se mu=cos 6, 
u—-Nu du 
(7) ¥= [lee /U 
N-wu du I 1 
= e +N aes 
(8) ¢ [ars ue (< x) 
(9) A?2?U =(1 —27) (A— N?- 2A Pu) — (2 — Nze)? 
=(1-w?)(h—2? - 2APu)— (N—nu)?. 
Here we must make the criticism that until p. 299 the 
mechanical interpretation of the quantities P, 2, N is 
| not very clearly defined so as to have their numerical 
values assigned in the C.G.S. system of units. Given the 
top we are to experiment with, we must first weigh it ; 
denote the weight by W grammes ; next measure the 
distance, 4 cm., between the C.G. and the point of 
support ; then Wg, dyne-cm., is the static moment 
denoted here by P, g denoting the acceleration of 
gravity (981) in cm./s.2. The number A denotes, as 
usual, the moment of inertia, in g.cm.%, of the top 
about an axis through its point perpendicular to its axis 
of figure ; A can be determined experimentally by swing- 
ing the top as a plane or conical pendulum, and measur- 
ing the length of the equivalent pendulum, / cm., and the 
angular velocity #z when swung without rotation as a 
conical pendulum of small aperture ; then 
