206 REPORT—1862. 
theorem of the Ultimate Multiplier. It may be noticed that in the last-men- 
tioned case the force-function is of the form =? so that if we represent also 
the eentral force by means of a force-function R (=function of r), then the 
entire foree-function is aot The case is a very interesting one; it in- 
cludes that considered § iv. of Bertrand’s “ Mémoire sur les équations différen- 
tielles de la Mécanique” (1852), where the force-function is of the form =5 
Motion of three mutually attracting bodies in a right line. 
88. The problem is considered by Euler in the memoir “De Motu rectilineo 
&e,” (1765), the forces being as the inverse square of the distance; and a 
solution is obtained for an interesting particular case. Let A, B,C be the 
masses, and suppose that at the commencement of the motion the distances 
CB, BA are in the ratio a: 1, and that the velocities (assumed to be in the 
same sense) are proportional to the distances from a fixed point. Then, if 
be the real root (there is only one) of the equation of the fifth order 
C0 (14+8a+3a7)=Ad’ (a’4+38a+3)+B(a+1*y(a’—1), 
the distances CB, BA will always continue in the ratio a:1. It may be 
added that the distances CB, BA each of them vary as *—a*, where a is a 
constant, and 7 is, according to the initial circumstances, a function of t de- 
fined by one or the other of the two equations 
r+ Nee 
a BI 
t=n'e¥ P—@—n'a? log 
a) aoa til 
t=n3eV 2—r?+n'a7sin— -. 
a 
89. The bodies are considered as restricted to move in a given line; but it 
is clear that if the bodies, considered as free points in space, are initially in a 
line, and the initial velocities are also in this line, then the bodies will always 
continue in this line, which will be a fixed line in space. But if the distances 
and velocities are as above, except only that the velocities, instead of being 
along the line, are parallel to each other in any direction whatever, then the 
bodies will always continue in a line, which is in this case a moveable line 
in space (see post, No. 93). 
90, Euler resumes the problem in the memoir of 1776 in the ‘ Nova Acta 
Petrop” The distances AB, BC being p and gq, then 
PP. pre Ny Be at Gus 
dO pg? tng)” 
CE SPU PAE) Gt 0 
and in particular he considers the before-mentioned case of a solution of the 
form p=nqg; and also the particular problem where one of the masses 
vanishes, C=0; in this case, introducing (instead of p, q) the new variables 
u, 8, Where g=up, dq=sdp (a transformation suggested by the homogeneity — 
of the equations), and making, moreover, the particular supposition that the 
integral of the first equation is (Zy—-—= (viz, making the constant 
