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 anywhere 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 the 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 geometrical 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 Euler had said something on the subject in the [Berlin] Memoirs for 1749 and 1750 (post, No. 167), and had constructed a new mechanical principle, but without pursuing the question. Segner's course of investigation 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 x, y, z such that fyzdm=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 fXYdm=0, XZdm=0, cos a cos B viz. the unknown quantities being taken to be t= ,T= (α, B, Y, cos y' cos Y being the inclinations of the principal axis to those of x, y, z), and then putting A=fa'dm, &c. (F=0 by hypothesis), Segner's equations for the determination of t, r are

G't2+(C'—A') t—G'—H'r=0,
(C'-B') r-G'tr+H't=0,

the second of which gives

T=

H't
'B'—C'+G't'

and by means of it the first gives

and

G'2ƒ3 —G'(A'—B')ť2+{(B'—C')(C'—A')—G'2—H'2}t+G' (B'—C')=0, which being a cubic equation shows that there are three principal axes; 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 u=fX'dm, we have

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 have been first given by Chasles, Note XXXI. of the 'Aperçu Historique' (1837).

by means of which Segner's equations may be verified. I have given this analysis, as the first solution of such a problem is a matter of interest.

148. There is little if anything added to Segner's results by the memoir, Euler, "Recherches sur la Connaissance Mécanique des Corps " (1758), which is introductory to the immediately following one on Rotation.

149. Relating to the theory of principal axes we have Binet's "Mémoire ́sur les Axes Conjugués," &c. (1813). The author proposes to make known the new systems of axes which he calls conjugate axes, which, when they are at right angles to each other, coincide with the principal axes; viz. considering the sum of the molecules each into its distance from a plane, such distance being measured in the direction of a line, then (the direction of the line being given) of all the planes which pass through a given point, there is one for which the sum in question is a minimum, and this plane is said to be conjugate to the given line, and from the notion of a line and conjugate plane he passes to that of a system of conjugate axes. The investigation (which is throughout an elegant one) is conducted analytically; the coordinates made use of are oblique ones, and the formulæ are thus rendered more complicated than they would otherwise have been; in referring to them it will be convenient to make the axes rectangular.

150. One of the results is the well-known equation

(A'—0)(B'—0)(C'—0)—F'2(A'—0)—G'2(B'—0)—H'2(C'—0)+2F'G'H'=0; which, if a1, y, z, are the principal axes, has for its roots fa,dm, fy,dm, Szidm.

x,

And the equations (1), p. 49, taking therein the original axes as rectangular, are

where A', ', E, F, G, H' denote the reciprocal coefficients — B'C' — F122

&c., and K' is the discriminant

A'B'C'—A'F'2—B'G”—C'H"+2FG'H': this is a symmetrical system of equations for finding cos a: cos ẞ: 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 Ox, Oy, Oz are the principal axes through a point the coordinates whereof are f, g, h, and if '=(say) fxdm, then we have

(in which I have restored the mass M, which is put equal to unity), so that if → have a given constant value, the locus of the point is a quadric surface, the nature whereof will depend on the value of 1. The surfaces in question are con202 y2 z2 1 focal with each other [and with the imaginary surface +. ·+. -A A' B

which is similar to the ellipsoid

x2 y2, z2 1
A' B'

=

-CM'

ABM which is the reciprocal of the comomental ellipsoid A'x2+B'y2+C'z2=Mk2 in regard to a concentric

sphere, radius k]. The author mentions the ellipsoid + +

x2 y2, z2 1
A' B' C' M

=

(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 incidentally considered is not the comomental ellipsoid itself, but, as just remarked, 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. Ampère's Memoir (1823).-The expression permanent axis 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 z2 for the centre of gravity.

the ellipsoid of gyration (+7 +7 = 11)

++
A B C M

154. Cauchy, in the memoir "Sur les Momens d'Inertie" (1827), considers the momental ellipsoid (A, B, C, F, G, HXx, y, z)2=1, and employs it as well to prove the existence of the principal axes as to determine their direction, and also the magnitudes of the principal moments; the results are obtained in the simplest and best forms; viz. the direction cosines are given by

(A—0)(B—0)(C—0)—(A—0) F2—(B—0) G2—(C—0) H2+2FGH=0,

→ being one of the principal moments.

155. Poinsot, "Mémoire sur la Rotation" (1834), defines the "Central

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 diameter. 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 momental ellipsoid Ax2+By2+Cz2=MZ+; and moreover that Poinsôt 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 principal axes; viz. these are

(A'—ə')(B'—ə')(C'—ə')—(A'—ə') F'2 — (B'—ə') G'2 —(C'—0') F'2+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, F, G, HXx, y, z)2=Mk

(A, B, C, F, G, H ut suprà), 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 afterwards, referring the ellipsoid to its principal axes, so that the equation is Ax2+By+C2=M, he writes A-Ma2, B=Mb2, C=Mc2, which reduces the equation to a2x2+b2y2+c2z2=k, and he considers the reciprocal ellipsoid x2, y2 22 x2, y2, z2 1 ++=1, or, what is the same thing,+ which is the ellip+ A B CM' soid of gyration.

2

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, HYx, y, z)2=Mk';

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', B', C', F', G', H'(x, y, z)=M (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 x2, y2 z2 + for the centre of gravity. And for B CM

ellipsoid of gyration (

determining the moments of inertia at the given point (say its coordinates are, n,) he obtains the equation

where the three roots of the cubic in P are the required moments. Analytically nothing can be more elegant, but, as already remarked, a geometrical construction for the magnitudes of these moments appears to be required.

160. Some very interesting geometrical results are obtained by considering the "equimomental surface" the locus of the points, for which one of the moments of inertia is equal to a given quantity II; the equation is of

course

and which includes Fresnel's wave-surface. In particular it is shown that the equimomental surface cuts any surface

confocal with the ellipsoid of gyration in a spherical conic and a curve of curvature; a theorem which is also demonstrated, Cayley, "Note on a Geometrical Theorem, &c." (1846).

161. Townsend, "On Principal Axes, &c." (1846).-This elaborate paper is contemporaneous, or nearly so, with Thomson's, and several of the conclusions are common to the two. From the character of the paper, I find it difficult to give an account of it; and I remark that, the theory of principal axes once brought into connexion with that of confocal surfaces, all ulterior developments belong more properly to the latter theory.

162. Haton de la Goupillière's two memoirs, "Sur la Théorie Nouvelle de la Géométrie des Masses" (1858), relate in a great measure to the theory of the integral sxydm, and its variations according to the different positions of the two planes x=0 and y=0; the geometrical interpretations of the several results appear to be given with much care and completeness, but I have not examined them in detail. The author refers to the researches of Thomson

and Townsend.

163. I had intended to show (but the paper has not been completed for publication) how the momental ellipsoid for any point of the body may be obtained from that for the centre of gravity by a construction depending on the "square potency" of a point in regard to the last-mentioned ellipsoid.

The Rotation of a solid body. Article Nos. 164–207.

164. It will be recollected that the problem is the same for a body rotating about a fixed point, and for the rotation of a free body about the centre of gravity; the case considered is that of a body not acted on by any forces. According to the ordinary analytical mode of treatment, the problem depends upon Euler's equations

Adp+(C-B) grdt=0,

Bảq+(A−C) rpdt=0,
Cdr+(B-A)pqdt=0,

for the determination of p, q, r, the angular velocities about the principal