ON THE SPECIAL PROBLEMS OF DYNAMICS. 227 
&e., and K’ is the discriminant =A'B'C’—A'F?—B'G?—C'H?+2FG'H : 
this is a symmetrical system of equations for finding cos « : cos GB: cos y, 
less simple however than the modern form (post, No. 154), the identity of 
which with Binet’s may be shown without difficulty. 
151. Another result (p. 57) is that if the original axes are principal axes, 
and if Ow, Oy, Oz are the principal axes through a point the coordinates 
whereof are f, g, h, and if ©,'= (say) ef x,*dm, then we have 
Mie ae UY ee A 
6,/—A''6,/—B 1 6/—0 Mf 
(in which T have restored the mass M, which is put equal to unity), so that if 
0,’ have a given constant value, the locus of the point is a quadric surface, the 
nature whereof will depend on the value of 6,. The surfaces in question are con- 
ig ze i! 
focal with each other [and with the imaginary surface =a a 
a 
2 2 
which is similar to the ellipsoid +h to=e which is the reciprocal of 
the comomental ellipsoid A'a?+B'y?+C'z?=Mz#* in regard to a concentric 
2 2 2 
sphere, radius /]. The author mentions the ellipsoid vt+e + a =F (see p. 64), 
and he remarks that his conjugate axes are in fact conjugate axes in respect 
to this ellipsoid, and consequently that the principal axes are in direction the 
principal axes of this ellipsoid: it is noticeable that the ellipsoid thus inci- 
dentally considered is not the comomental ellipsoid itself, but, as just re- 
marked, its reciprocal in regard to a concentric sphere. 
152. Poisson, ‘ Mécanique’ (1st ed. 1811, and indeed 2nd ed. 1833), gives 
the theory of principal axes in a less complete form than in Binet’s memoir; 
for the directions of the principal axes are obtained in anything but an 
elegant form. 
153. Ampére’s Memoir (1823).—The expression permanent awis is used 
in the place of principal axis, which is employed to designate a principal 
axis through the centre of gravity. The memoir contains a variety of very 
interesting geometrical theorems, which however, as no ellipsoid is made use of, 
can hardly be considered as exhibited in their proper connexion. The author 
arrives incidentally at certain conics, which are in fact the focal conics of 
‘peo eek 
pt oru) | 
154, Cauchy, in the memoir “Sur les Momens d’Inertie ” (1827), considers 
the momental ellipsoid (A, By eG: HY, y, z)=1, and employs it as 
well to prove the existence of the principal axes as to determine their di- 
rection, and also the magnitvdes of the principal moments; the results are 
a in the simplest and best forms; viz. the direction cosines are given 
7 
the ellipsoid of gyration (G+ for the centre of gravity. 
(A—@) cosa +H cos B+G cos y=0, 
H cos a-+(B—6) cos B+F cos y=0, 
G cosa+F cos 3-+(C—6) cos y=0, 
where 
(A—0)(B—0e)(C—0)—(A—6) F’—(B—@) G?—(C—6) H’4+ 2FGH=0, 
© being one of the principal moments. 
155. Poinsot, “Mémoire sur la Rotation” (1834), defines the “Central 
a2 
