224. REPORT—1862. 
But this, the original form of the theorem, is a mere deduction from a general 
theory of the representation of the integrals 
A xdm, ay ydm, fzdm, Syed, fzxdm, fp aydm 
for any axes through the given origin by means of an ellipsoid depending on 
the values of these integrals corresponding to a given set of rectangular axes 
through the same origin. 
143. If, for convenience, we write as follows, M= f dm the mass of the 
body, and 
A! = fxd, B’ =fy'dm, C=fzdm, EF’ =f yzdm, G’ =faxdm, H’ =f xydm, 
and moreover 
A=f(y’+#) dm, B=/{(@ +") dm, O=/(2*+y’) dm, 
=—fyzdm, cc —fzxdm, H=—/f«ydm*, 
so that 
A=B'+C', B=C'+A', C=A'4+ BY, F=—F’, G=—G', H= 3) 
then the ellipsoid which in the first instance presents itself for this purpose, 
and which Prof. Price has termed the Ellipsoid of Principal Axes, but which 
IT would rather term the “‘ Comomental Ellipsoid,” is the ellipsoid 
(A’, BY, C, F’, G’, H (a, y, z=) =Mk, 
where k is arbitrary, so that the absolute magnitude is not determined. But 
it is more usual, and in some respects better to consider in place thereof the 
« Momental Ellipsoid” (Cauchy, ‘Sur les Moments d’Inertie,” Exercices de 
Mathématique, t. ii. pp. 93-103, 1827), 
(A, B, C, F, G, HYa, y, 2) =Mht, 
or as it may also be written, 
(A'4+B40)aety+e)—(A4 B, C’, F, Gg, H'{«, Y; z) =MM, 
which shows that the two ellipsoids have their axes, and also their circular 
sections coincident in direction. 
144. And there is besides this a third ellipsoid, the ‘ Ellipsoid of Gyra- 
tion,” which is the reciprocal of the momental ellipsoid in regard to the con- 
centric sphere, radius &. The last-mentioned ellipsoid is given in magnitude, 
viz., if the body is referred to its principal axes, then putting A>=Ma’*, B= M0”, 
C=Mc’, the equation of the ellipsoid of gyration is 
2 2 2 
wv y z 
lee ee a ee) | 
—+ats 
The axes of any one of the foregoing ellipsoids coincide in direction with the 
principal axes of the body, and the magnitudes of the axes lead very simply 
to the values of the principal moments A, B, C. 
145. The origin has so far been left arbitrary: in the dynamical applica- 
tions, this origin is in the case of a solid body rotating about a fixed point, 
the fixed point; and in the case of a free body, the centre of gravity. But 
the values of the coefficients (A, B, C, F, G, H), or (A7 3’, OC 2 Ga 
corresponding to any given origin whatever, are very easily expressed in 
* | have ventured to make this change instead of writing as usual F= f' yzdm, &e.; asin — 
most cases F=G=H=0, the formule affected by the alteration are not numerous. 
P44 2 
