ON THE SPECIAL PROBLEMS OF DYNAMICS. 225 
terms of the coordinates of this origin, and the values of the corresponding 
coefficients for the centre of gravity as origin; or, what is the same thing, 
any one of the ellipsoids for the given origin may be geometrically constructed 
by means of the ellipsoid for the centre of gravity. The geometrical theory, 
as regards the magnitudes of the axes, does not appear to have been any- 
where explicitly enunciated; as regards their direction, it is comprised in the 
theorem that the directions at any point are the three rectangular directions 
at that point in regard to the ellipsoid of gyration for fhe centre of gravity*, 
post, No. 159. The notion of the ellipsoids, and of the relation between the 
ellipsoids at a given point and those at the centre of gravity, once established, 
the theory of principal axes and moments of inertia becomes a purely geo- 
metrical one. 
146. The existence of principal axes was first established by Segner in the 
work ‘Specimen Theorie Turbinum,’ Halle (1755), where, however, it is 
remarked that Kuler had said something on the subject in the [Berlin] Me- 
moirs for 1749 and 1750 (post, No. 167), and had constructed a new mecha- 
nical principle, but without pursuing the question. Segner’s course of inves- 
tigation is in principle the same as that now made use of, viz. a principal axis 
is defined to be an axis, such that when a body revolves round it the forces 
arising from the rotation have no tendency to alter the position of the axes. 
It is first shown that there are systems of axes a, y, z such that of yzdm =0, 
and then, in reference to such a set of axes, the position of a principal axis, 
say the axis of X, is determined by the conditions oh: XYdnm=0, wh XLdm=0, 
cos a cos 
viz. the unknown quantities being taken to be t=——, -= (a, 3, y> 
cos y cos y 
being the inclinations of the principal axis to those of a, y, z), and then 
putting A= =f x’dm, &c. (F=0 by hypothesis), Segner’s equations for the de- 
termination of t¢, 7 are ; 
G'?+(C'—A’) i—G'—H'r=0, 
(C'—B') r—G'tr+Ht=0, 
the second of which gives 
scr URES 
SS yang 
and by means of it the first gives 
G?—G'(A'’—B')?+ {(B’—C')(C'—A')—G?—H} + G' (B'—C') =0, 
which being a cubic equation shows that there are three principal axes; and 
it is afterwards proved that these are at right angles to each other. 
147. To show the equivalence of Segner’s solution to the modern one, I 
remark that if w= f° X?dm, we have 
(Nowra oe 
B.. t+@—wrtF =0, 
Gig che cba an Ayes 
whence 
* The rectangular directions at a point in regard to an ellipsoid are the directions of 
the axes of the circumscribed cone, or, what is the same thing, they are the directions of 
the normals to the three quadric surfaces confocal with the given ellipsoid, which pass 
through the given point. ‘The theory of confocal surfaces appears to haye been first given 
by ay Note XXXTI. of the ‘ Apergu Historique’ (1837). 
- Q 
