230 REPORT—1862. 
axes; considering p, g, 7 as known, we obtain by merely geometrical consi- 
derations 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 cireum- 
stances 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 formule 
‘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 deve- 
lopments of Rueb and Jacobi; the method of Poinsot is also based upon the 
consideration of the invariable plane. 
166. D’Alembert’s principle, which affords a direct and general method 
for obtaining the equations of motion in any dynamical problem whatever, 
was given in his “ Traité de Dynamique”’ (1743); and in his memoir of 1749 
he applied it to the physical problem of the Precession of the Equinoxes, which 
is a special case of the problem of Rotation, the motion of rotation about the 
centre of gravity being in fact similar to that about a fixed point. But, as 
might be expected in the first attempt at the analytical treatment of so 
difficult a problem, the equations of motion are obtained in a cumbrous and 
unmanageable form. 
_ 167. They are obtained by Euler in the memoir “ Découverte d’un Nou- 
veau Principe de Mécanique,” Berlin Memoirs for 1750 (1752) (written 
before the establishment of the theory of principal axes), in a perfectly 
elegant form, including the ordinary one already mentioned, and, in fact, 
reducible to it by merely putting the quantities F, G, H (which denote the 
integrals JI yzdm, &c.) equal to zero. But Euler does not in this memoir 
enter into the question of the integration of the equations, 
168. The notion of principal axes having been suggested by Euler, and 
their existence demonstrated by Segner, we come to Euler’s investigations 
contained in the memoirs “‘ Du Mouvement de Rotation &c.,” Berlin Me- 
moirs for 1758 (printed 1765) and for 1760 (printed 1767), and the “ Theoria 
Motus Corporum Solidorum &c.” (1765). In the memoir of 1760, and in 
the “ Theoria Motus,” Euler employs s, the angular velocity round the in- 
stantaneous axis, but not the resolved velocities & cos a, & cos 3,8 cos y 
(=p, 4, 7): these quantities (there called w, y, z) are however employed in 
the memoir, Berlin Memoirs (1758), which must, I apprehend, have been 
written after the other, and in which at any rate the solution is developed. 
with much greater completeness. It is in fact carried further than the 
ordinary solutions, and after the angular velocities p, g, r have been found, 
the remaining investigation for the position in space of the principal axes, 
conducted, as above remarked, without the aid of the invariable plane, is one 
of great elegance. 
169. In the last-mentioned memoir Euler starts from the equations given 
by d’Alembert’s principle ; viz. the impressed forces being put equal to zero, 
these are 
dm (ve ‘at)= 0, &e., 
or, what is the same thing, using u,v, w to denote the velocities of an element 
in the directions of the axes fixed in space, these are 
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