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REPORT ON THE PROGRESS OE THE SOLUTION OF 
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161. Townsend, “ On principal Axes &c.” (1846).—This elaborate paper is con 
temporaneous, 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 Goupilliere’s two memoirs, “ Sur la Théorie Nouvelle de la 
Géométrie des Masses” (1858), relate in a great measure to the theory of the integral 
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 publi 
cation) 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 
A dp + (G— B) qrdt = 0, 
Bdq + (A — C) rpdt = 0, 
Gdr + (B — A ) pqdt = 0, 
for the determination of p, q, r, the angular velocities about the principal axes; con 
sidering p, q, r as known, we obtain by merely geometrical considerations a system of 
three differential equations of the first order for the determination of the position in 
space of the principal axes. 
165. The solution of these, which constitutes the chief difficulty of the problem, 
is usually effected by referring the body to a set of axes fixed in space, the position 
whereof is not arbitrary, but depends on the initial circumstances of the motion; viz. 
the axis of z is taken to be perpendicular to the so-called invariable plane. The 
solution is obtained without this assumption both by Euler and Lagrange, although, as 
remarked by them, the formulae are greatly simplified by making it; it is, on the 
other hand, made in the solution (which may be considered as the received one) by 
Poisson; and the results depending on it are the starting-point of the ulterior analytical 
developments of Rueb and Jacobi; the method of Poinsot is also based upon the con 
sideration of the invariable plane.