ON THE SPECIAL PROBLEMS OF DYNAMICS. 207 
of integration to vanish), he obtains between s and w the equation of the first 
order 
ds A B 
which, however, he is not able to integrate. 
91. Jacobi has given in the memoir “ Theoria Noyi Multiplicatoris ” (1845) 
($28, entitled “De Problemate trium corporum in eadem recta motorum. Sub- 
stitutio Euleriana. Theoremata de viribus homogeneis’’) a very symmetrical 
and elegant investigation of the same problem, The centre of gravity being 
assumed to be at rest, the coordinates «, a,, x, of the three bodies are in the first 
instance expressed as linear functions of the two variables u, v (being, as Jacobi 
remarks, the transformation employed in his memoir “Sur I’élimination des 
2, 2 
Neeuds” (1843), post, No. 114), CY and - come out respectively equal to 
a au 
homogeneous functions of the degree —2 of these variables u and v, and the 
integral of Vis Viva exists, The subsequent transformation consists in the 
introduction of the variables 7, $, 8, n, Where w=r cos $, v=r sing, s= V7 ees 5 
r 
7= VP Se this gives a system of equations independent of x; viz., 
dg: ds; dn=n: 38°+n°?—®: —3sn+@', 
where @ is a given function of ¢, and ©’ is the derived function. If these 
equations were integrated, the equation of Vis Viva gives at once r= 
; 2 (®—2(s*+n°)); and finally the time ¢ would be given by a quadrature, 
1 
; V o—3(s°+ 7° 
if one integral were known the other would be at once furnished by the 
general theory, There is a simplification in the form of the solution if h (the 
constant of Vis Viva) =0. It is remarked that the method is equally appli- 
cable when the force varies as any power of the distance; and moreover that 
when the force varies as (dist.)~*, then the solution.depends upon one qua- 
drature only. 
92, The concluding part of the section relates to the very general problem 
of a system of n particles acted on by any forces homogeneous functions of 
the coordinates (this includes the case of n particles mutually attracting each 
other according to a power of the distance), and this more general investiga- 
tion illustrates the method employed in regard to the three bodies in a line. 
Tt may be remarked that in the general theorem for the » particles “sint 
vires &c.,” the constant of Vis Viva is supposed to vanish. 
The system of three equations has the multiplier M= hehte 
Particular cases of the motion of three bodies. 
93. In the case of three bodies attracting each other according to the in- 
verse square of the distance, the bodies may move in such manner as to be 
constantly in a line (a moveable line in space); this appears by the memoir, 
Euler, “Considérations générales, &c.” (17 64), in which memoir, however 
(which it will be observed precedes the memoir De Motu rectilineo &c.” 
(1765), referred to No. 88), Euler assumes that the mass of one of the bodies 
is so small as not to affect the relative motion of the other two. Calling 
the bodies the Sun, Earth, and Moon, and taking the masses to be 1, m, 0, 
then a result obtained is, that in order that the Moon may be perpetually 
