232 REPORT—1862. 
The formule, although complicated, are extremely elegant, and they appear 
to have been altogether overlooked by subsequent writers. 
171. Euler remarks, however, that the complexity of his solution is owing 
to the circumstance that the fixed point P is left arbitrary, and that they 
may be simplified by taking this point so as that a certain relation G—3B°=0 
may be satisfied between the constants of the solution; and he gives the far 
more simple formule corresponding to this assumption. This amounts to 
taking the point P in the normal of the invariable plane, and the resulting 
formule are in fact identical with the ordinary formule for the solution of 
the problem. The expression invariable plane is not used by Euler, and 
seems to have been first employed in Lagrange’s memoir “ Essai sur le Pro- 
bléme de Trois Corps,” Prix de l’Acad. de Berlin, t. ix. (1772): the theory 
in reference to the solar system has been studied by Laplace, Poinsot, and 
others. 
172. Lagrange’s solution in the memoir of 1773 is substantially the same 
with that in the ‘Mécanique Analytique.’ Only he starts from the integral 
equations of areas and of Vis Viva, but in the last-mentioned work from the 
equations of motion as expressed in the Lagrangian form by means of the 
Vis Viva function T (=23(«?+y"+2")dm). The distinctive feature is that 
he does not refer the body to the principal axes but to any rectangular 
axes whatever fixed in the body: the expression for T consequently is 
T=3(A,B,C,F,G, HY, q,7)*, instead of the more simple form 
T=3(Ap’+ Bqg’+Cr’), 
which it assumes when the body is referred to its principal axes. And 
Lagrange effects the integration as well with this more general form of T, as 
without the simplification afforded by the invariable plane; the employment, 
however, of the more general form of T seems an unnecessary complication 
of the problem, and the formule are not worked out nearly so completely as 
in Euler’s memoir. It may be observed that p, ¢, 7 are expressed as functions 
of the instantaneous velocity w(—= p?+q°+7"), and thence ¢ obtained by a 
quadrature as a function of w. 
173. Poisson’s Memoir of 1809.—The problem is only treated incidentally 
for the sake of obtaining the expressions for the variations of the arbitrary 
constants ; the results (depending, as already remarked, on the consideration 
of the invariable plane) are obtained and exhibited in a very compact form, 
and they have served as a basis for further developments ; it will be proper 
to refer to them somewhat particularly. The Eulerian equations give, in the 
first place, the integrals 
Ap? + Bq? +Cr* =h, 
Ap? + B+ Crs’ ; 
and then by means of these, », g being expressed in terms of 7, we have ¢ in 
terms of r by a quadrature. 
174. The position in space of the principal axes is determined by referring 
them, by means of the angles 6, ¢, c, to axes Ow, Oy, Oz fixed in space ; if, to 
fix the ideas, we call the plane of wy the ecliptic (Ox being the origin of 
longitudes), and the plane of the two principal axes «, y, the equator, then we 
have 
0, the longitude of node, 
g, the inclination, 
c, the hour-angle, or angular distance of Ox, from the node, 
and a, 3, y the cosine inclinations of Ow,, a’, 6’, y' those of Oy,, and a”, 6", y” 
