228 REPORT—1862. 
Ellipsoid’’ as an ellipsoid having for its axes the principal axes through the 
centre of gravity, the squares of the lengths being reciprocally proportional 
to the principal moments; and he remarks in passing that the moment about 
any diameter of the ellipsoid is inversely proportional to the square of this dia- 
meter. It is to be noticed that Poinsot speaks only of the ellipsoid having 
its centre at the centre of gravity, but that such ellipsoid may be constructed 
about any point whatever as centre, so generalized, it is in fact the mo- 
mental ellipsoid Aw*+By’+Cz=Mk*; and moreover that Poinsot defines 
his ellipsoid by reference to the principal axes. 
156. Pine, “On the Principal Axes, &c.” (1837), obtained analytically in 
a very elegant manner equations for determining the positions of the prin- 
cipal axes; viz. these are 
(A'—0') cosa +H’ cos 3-+G’ cos y=0, 
H’ cos a+(B'—60’') cos B+ FE" cos y=0, 
Gs cosa+F" cos 8 +(C'—0’) cos y=0, 
where 
(A'—0')(B'—0')(C'— 9')—(A'— 0’) F?— (B'—0') G?—(C'— 0) F? ++ 2F'G'H'=0; 
viz. these are similar to those of Cauchy, only they belong to the comomental 
instead of the momental ellipsoid. 
157. Maccullagh, in his Lectures of 1844 (see Haughton), considers the 
momental ellipsoid 
(A, B,C) Hh, G; HY2, Ys 2) = Mk 
(A, B, C, F, G, H ut supra), which is such that the moment of inertia of the 
body with respect to any axis passing through the origin is proportional to 
the square of the radius vector of the ellipsoid; and from the geometrical 
theorem of the ellipsoid having principal axes he obtained the mechanical 
theorem of the existence of principal axes of the body; at least I infer that 
he did so, although the conclusion is not explicitly stated in Haughton’s 
account ; but in all this he had been anticipated by Cauchy. And after- 
wards, referring the ellipsoid to its principal axes, so that the equation is 
Aa’ + By’? +C2?=Mk", he writes A=Ma’*, B=Mé*, C=Me’, which reduces 
the equation to a°w*+ b°y*+¢2°=k*, and he considers the reciprocal ellipsoid 
2 2 w2 a2 2 aw 
atts= 1, or, what is the same thing, <+% $o=5 which is the ellip- 
soid of gyration. 
158. Thomson, ‘On the Principal Axes of a Solid Body” (1846), shows 
analytically that the principal axes coincide in direction with the axes of the 
momental ellipsoid 
(A, B, C, F, G, He, y, z) =Mi'; 
but the geometrical theorem might have been assumed: the investigation is 
really an investigation of the axes of this ellipsoid. And he remarks that 
the ellipsoid (A', Be Eas He, Ys z) =Mke (the comomental ellipsoid) 
might equally well have been used for the purpose. 
159. He obtains the before-mentioned theorem that the directions of the 
principal axes at any point are the rectangular directions in regard to the 
ene : fe ger aan | 
ellipsoid of gyration (G+ 3 +9-3) 
determining the moments of inertia at the given point (say its coordinates 
are £, n, ¢) he obtains the equation 
for the centre of gravity. And for 
