; 208 REPORT—1862. 
in conjunction, its distance must be to that of the Sun as @:1, where 
m(1—a)’=3a’—3a'+a’, or a= 3/ im nearly. It appears, however (ante, 
No. 88), that the foregoing restriction as to the masses is unnecessary, and, as 
will be mentioned, the problem has since been treated without such restriction. 
Euler investigates the motion in the case where the initial carcumstances are 
nearly but not exactly as originally supposed; this assumes, however, that 
the motion is stable—z.e. that the bodies will continue to moye nearly, but 
not exactly as originally supposed, which is at variance with the conclusions 
of Liouville’s memoir, post, No. 95. I have not examined the cause of this 
discrepancy. 
94, In the ‘Mécanique Céleste’ (1799), Book x. ¢. 6, Laplace considers 
two cases where the motion can be exactly determined. 
1°. Force varies as any function of the distance. It is shown that the 
motion may be such that the bodies form always an equilateral triangle of 
variable magnitude—the motion of each body about the centre of gravity 
being the same as if that point were a centre of force attracting the body 
according to a similar law. 
2°. Force qx (dist.)". The motion may be such that the three bodies are 
always in a right line (moveable in space), the relative distances being in 
fixed ratios to each other. In particular, if force q (dist.)~?, then 
m, m', m' being the masses, the quantity z which determines the ratio of the 
distances mm’, m'm is given by 
O=mz?[(1+2)?—1]—m' (142) (1—2)—m" [(1+2)°—27]=0, 
which is, in fact, the formula in Euler’s memoir “ De Motu rectilineo &c.’’ 
95. Liouville’s memoir “Sur un cas particulier &c.” (1842) has for its 
object to show that if the initial circumstances are not precisely as supposed 
in the second of the two cases considered by Laplace, or, what is the same 
thing, in Euler’s memoir ‘ Considérations générales &c.,” then the motion is 
unstable ; the instability manifests itself in the usual manner, viz. the expres- 
sions for the deviations from the normal positions are found to contain real 
exponentials which increase indefinitely with the time. 
96. It may be proper to refer here to Jacobi’s theorem, ‘ Comptes Rendus,’ 
t. iii. p. 61 (1836), quoted in the foot-note p. 15 of my Report of 1857, 
which relates to the motion of a point without mass revolving round the Sun, 
and disturbed by a planet moving in a circular orbit, and properly belongs (as 
I have there remarked) to the problem of two centres, one of them moveable 
and the other revolying round it in a circle with uniform velocity. The 
theorem (given without demonstration by Jacobi) is proved by Liouville in 
his last-mentioned memoir, and he remarks that the theorem follows very 
simply as a corollary of the theorem by Coriolis, “On the Principle of Vis Viva 
in Relative Motions,” Journ. de l’Ecole Polyt. t. xiii. p. 268 (1832). There 
is, however, no difficulty in proving the theorem; another proof is given, 
Cayley, “ Note on a Theorem of Jacobi’s &c.” (1862). 
Motion in a resisting medium. 
97. I do not consider the various integrable cases of the motion of a par- 
ticle in a resisting medium, the resistance varying with the velocity according 
to some assumed law, the particle being either not acted on by any force, or 
acted upon by gravity only. Some interesting cases are considered in Jacobi’s 
memoir “ De Motu puncti singularis” (1842), §§ 6 and 7 (see post, No. 108). 
