ON THE SPECIAL PROBLEMS OF DYNAMICS. | 217 
du_du,_dv_dv,_dw_dw,_dr_dr, | 
UU,” V*~ Vi AeW.gb Wiyb Rowe, 
where U, U,, &c. are functions of the quantities wu, u,, v, &c. Omitting from 
the system the equation (dt), there are eight equations between nine quan- 
tities; but there are two known integrals, viz., the integral of Vis Viva and 
the integral of principal moment (or sum of the squares of the integrals of 
areas); that is to say, the system is really a system of sta equations. 
119. Painyin, “Recherche du dernier Multiplicateur &e.” (1854).—The 
author investigates the ultimate multiplier for two systems of differential 
equations :— 
1°. The system of the equations I. to VI. in Jacobi’s memoir “Sur 
Vélimination des Neuds &c.” (anté, No. 114). Writing in the equations 
Ory 
Mizy (8b. 
in the form 
r,', and treating 7", 7,’ as new variables, the system may be written 
du_du,_ U_d,_dr_dr,__dr' 
Te Ugo Mer B ar Ro 
which, omitting the equation (=dt), is a system of seven equations be- 
tween eight variables; and it is for this form of the system that the value 
of M is determined, the result obtained being the simple and elegant one, 
__ sin 7 sin 7, 
eee an” 1 
fact the equation V. of the system in Jacobi’s form, so that it is really a 
system of sia equations (ante, No, 115). 
2°, The system secondly discussed is Bertrand’s system of nine equations 
(ante, No. 118). The multiplier M is obtained under four different forms, 
| 1 1 1 ; F 
M= en jae AZT Baa (1 do not stop to explain the notation), 
the last of them being referred to as a result announced by M. Bertrand in 
his course. But it is shown by M. Bour in the memoir next referred to (post, 
No. 122), that the multiplier for the system in question can be obtained in a 
very much more simple manner, almost without calculation. 
120. In connexion with Jacobi’s theory of the elimination of the Nodes, I 
may refer to the investigations ‘‘ Application to the Problem of three Bodies ”’ 
Nos. 84 to 96 of Donkin’s memoir ‘‘ On a Class of Differential Equations &e.” 
Part II. The author remarks that his differential equations No. 93 afford an 
example of the so-called elimination of the Nodes, quite different however (in 
that they are merely transformations of the original differential equations of 
the problem without any integrations) from that effected by Jacobi. 
121. It may be right to refer again in this place to the concluding part of 
§ 28 of Jacobi’s memoir “ Nova Theoria Multiplicatoris ”’ (anté, No. 92), as 
bearing on the problem of three bodies. 
122. Bour’s “ Mémoire sur le Probléme des Trois Corps” (1856).—The 
author remarks that Bertrand’s system of equations have lost the remarkable 
form and the properties which characterize the ordinary equations for the 
solution of a dynamical problem. But by selecting eight new variables, 
functions of Bertrand’s variables, the system may be brought back to the 
standard Hamiltonian form 
dr! 
4 R,’( Ar dt), 
The system of seven equations has an integral which is in 
or to the form adopted by M. Bour, of a partial differential equation 
