ON THE SPECIAL PROBLEMS OF DYNAMICS, 229 
Ee ae ee 
2 2 2 ASP 2 2 2 BSE 2 2) 72 C=P’ 
Str +o +e Ste tote On tO + 
where the three roots of the cubic in P are the required moments. Analyti- 
cally 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 consider- 
ing the “equimomental surface” the locus of the points, for which one of 
the moments of inertia is equal to a given quantity IT; the equation is of 
course 
x 4 yy a 2 1 
ie oA a See 
ety topo" Pry tepoo™ ee 
and which includes Fresnel’s wave-surface, In particular it is shown that 
the equimomental surface cuts any surface 
i a ee 
A+6'B+0'C+o-M 
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 Geo- 
metrical 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 deye- 
lopments belong more properly to the latter theory. 
162. Haton de la Goupilliére’s two memoirs, “Sur la Théorie Nouvelle de 
la Géométrie des Masses” (1858), relate in a great measure to the theory of 
the integral of wydm, 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. Itwill 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, 
Bdq+ (A—C) rpdt=0, 
Cdr + (B—A) pqdt=0, 
for the determination of p, g,7, the angular velocities about the principal 
