ON THE SPECIAL PROBLEMS OF DYNAMICS. 197 
49. If the attractive force of one of the centres is taken equal to zero, 
then the position of such centre is arbitrary, and it may be assumed that the 
centre lies on the curve, which is in this case an ellipse (conic section) ; the 
expression of the time presents itself as a function of the focal radius vectors 
and the chord of the arc described ; which, as remarked, anté, No. 20, leads to 
Lambert’s theorem for elliptic motion. 
50. The case presents itself of an ellipse or hyperbola described under the 
: ; k dp arts ’ 
action of the two forces, viz. the equation VE WS will be satisfied 
byr—a=0, if r—a@ be a double factor of R, or by s—f@=0, if s—B be a 
double factor of S, a case which is also considered in the ‘ Mécanique Ana- 
lytique ;’ and see in regard to the analytical theory, t. ii. 3rd ed. Note III. by 
M. Serret, and “‘Thése,” Liouv. 1848. It is remarked by M. Bonnet, Note LV. 
and Liouy. t. ix. p. 113, 1844, that the result is a mere corollary of a general 
theorem, which is in. effect as follows, viz. if a particle under the separate 
actions of the forces F, F’, . . . starting in each case from the same point in 
the same direction but with the initial velocities v, v', &e. respectively, 
describe the same curve, then such curve will also be described under the 
conjoint action of all the forces, provided the body start from the same point 
in the same direction, with the initial velocity V= /v?+v7+4.. 
51. Lagrange’s second memoir (same volume of the Turin Memoirs) 
contains an exceedingly interesting discussion as to the laws of force for 
which the problem can be solved. Writing U,V, u,v in the place of Lagrange’s 
P, Q, p, g, the equations of motion are 
x, (wx—a)U 4 (a— a)V_ 0, 
dt? u v 
Uy , (y—5)U | (y—6)V_ 
Ger Sagem et Te 
dz, (z—c)U  (z—y)V_ 
de Tiare) Babies wih Int 
where 
w=  (x—a)’+(y—b)'+ (zc), 
v= (w7—a)’+(y—B) + (e—y)’, 
and putting also f (= /(a—a)’+ (6— 8)’ + (c—y)’) the distance of the centres, 
and then w’=/x, v’=fy, eet (~,y are of course not to be con- 
founded with the coordinates originally so represented), Lagrange obtains 
the equations 
Pax (w+y—D¥ 
poet ke SO +f (Xde+Ydy)=0, 
& —1)x 
Aces eee +f (Xdu+Ydy)=0, 
which he represents by 
Px 
gp tM=0, 
iy 
3 de +N=0; 
