[43 
27 G 
43. 
ON THE DIFFERENTIAL EQUATIONS WHICH OCCUR IN 
DYNAMICAL PROBLEMS. 
[From the Cambridge and Dublin Mathematical Journal, vol. n. (1847), pp. 210—219.] 
•Jacobi, in a very elaborate memoir, “ Theoria novi multiplicatoris systemati aequa 
tionum differentialium vulgarium applicandi ” ( J ), has demonstrated a remarkable property 
of an extensive class of differential equations, namely, that when all the integrals of 
the system except a single one are known, the remaining integral can always be deter 
mined by a quadrature. Included in the class in question are, as Jacobi proceeds to 
show, the differential equations corresponding to any dynamical problem in which 
neither the forces nor the equations of condition involve the velocities; i. e. in all 
ordinary dynamical problems, when all the integrals but one are known, the remaining 
integral can be determined by quadratures. In the case where the forces and equations 
of condition are likewise independent of the time, it is immediately seen that the 
system may be transformed into a system in which the number of equations is less 
by unity than in the original one, and which does not involve the time, which may 
afterwards be determined by a quadrature 1 2 ; and, Jacobi’s theorem applying to this new 
system, he arrives at the proposition “ In any dynamical problem where the forces 
and equations of condition contain only the coordinates of the different points of the 
system, when all the integrals but two are determined, the remaining integrals may 
be found by quadratures only.” In the following paper, which contains the demonstra 
tions of these propositions, the analysis employed by Jacobi has been considerably 
varied in the details, but the leading features of it are preserved. 
1 Crelle, t. xxvn. [1844], pp. 199—268 and t. xxix. [1845], pp. 213—279 and 333—376. Compare also the 
memoir in Liouville, t. x. [1845], pp. 337—346. 
- For, representing the velocities by x\ y' ... the dynamical system takes the form 
dt : dx : dy ... : dx' : dy' ... = 1 : x' : y' ... : X : Y ... , 
and the system in question is simply dx : dy ... : dx' : dy' ... = x' : y' ... : X : Y ... .