ON THE SPECIAL PROBLEMS OF DYNAMICS, 241 
fixed centre. The section is in fact headed “Solutio simultanea “Gara 
de motu puncti versus centrum attracti atque problematis de rotatione &c.” 
and Jacobi, after noticing that Poisson, in his memoir of 1816 (Mém. de 
l’Inst. t.i.), had shown that the expressions for the variations of the elements 
in the two problems could be investigated by a common analysis, remarks, 
«Sed ipsa problemata duo imperturbata hic primum, quantum credo, amplexus 
sum.” The solution is in fact as follows:—Suppose that in the one problem 
the position of the point in space, and in the other problem the position of 
the body in regard to the fixed axes is determined in any manner by the 
quantities 9,, 9,,9,- Let 
di , dq, di 
ly ag 
and expressing the Vis Viva function T in terms of ¢,, 9,5 Ys 1's Jo's Yq» let 
LT i BE rg AL 
dy? dg?” dg, 
and let H be the value of T expressed in terms of ¢,, 9,5 3s Py» Pos Py» 80 that 
H=a is the integral of Vis Viva (this is merely the transformation to the 
Hamiltonian form). And let H,=a,, ¢=«a,', p=a," be the three integrals 
of areas (H, H,, ¢, are functions of the variables only, not containing the 
arbitrary constants a, a,, a,',a,"). Then, expressing 
H, H,, H, (=VH'+9'+¥) 
in terms of p,, 75 Ps 91> Qo» Jo and by means of the equations 
H=7, a H.=¢; 
(where a,= Va,7+a,"+a,'") expressing p,, p,, p, in terms of 4, J.) %» We 
haye p,dq,+p,dq¢,+p,dq, a complete differential ; and putting 
§ (cae +p.dq, +p.ty,)=V, 
then (a, a,, a,, b, b,, 6, being arbitrary constants) we have 
H=a, H, =a, Hy =4,, 
a dp, 4 
Y((Ba TET G ries dq, + Pa aly, = t+, 
dV d dp 
ree ee Pea = Ps dy, + pe ay,)= it 
Les a dp, 4, \— 
r= ( (Pa da, + da, qg,+ da, a4,)=by 
as the complete fal: of either problem, the last three of them being the 
final integrals. 
And it is added that if in either problem we have H+. instead of H, the 
expressions for the variations of the elements assume the canonical forms 
da_ dQadb_ da 
dt db’ dt da’ 
The solution is not further developed as regards the rotation problem, but 
it is so (§ 67) as regards the other problem. 
ee It must, I think, be considered that a comprehensive memoir on the 
R 
=Ps) 
