26 REPORT—1857. 
48. Jacobi applies the theorem of the ultimate multiplier to the dif- 
ferential equations of dynamics, considered first in the unreduced La- 
grangian form, where the coordinates are connected by any given system of 
equations of condition; secondly, in the reduced or ordinary Lagrangian 
form ; and, thirdly, in the Hamiltonian form. The multiplier can be found 
for the first two forms, and the expressions obtained are simple and elegant ; 
but, as regards the third form, there is a further simplification : the multiplier 
M of the system is equal to the unity, and the multiplier of the last equation 
is therefore equal to V. The two cases are to be distinguished in which ¢ 
does not, or does enter into the equations of motion; in the latter case the 
theorem furnished by the principle of the ultimate multiple is the same as 
for the general case of a system, the multiplier of which is known, viz., the 
theorem is, given all the integrals except one, the remaining integral can be 
found by quadratures only. But in the former case, which is the ordinary 
one, including all the problems in which the condition of vis viva is 
satisfied, there is a further consequence deduced. In fact, the time ¢ may 
be separated from the other variables, and the system of differential equa- 
tions reduced to a system not involving the time, and containing a number 
of equations less by unity than the original system, and the theorem of the 
ultimate multiplier applies to this new system. But when the integrals of 
the new system have been obtained, the system may be completed by the 
addition of a single differential equation involving the time, and which is 
integrable by quadratures; the theorem consequently is, given all the 
integrals except two, these given integrals being independent of the time, 
the remaining integrals can be found by quadratures only. This is, in fact, 
the ‘ Principium generale mechanicum’ of the memoir of 1842. 
The last of the published writings of Jacobi on the subject of dynamics 
are the ‘ Auszug zweier Schreiben des Professors Jacobi an Herrn Director 
Hansen,’ Crelle, t. xlii. pp. 12-31 (1851): these relate chiefly to Hansen’s 
theory of ideal coordinates. 
49. The very interesting investigations contained in several memoirs by 
Liouville (‘ Liouville,’ t. xi. xii. and xiv., and the additions to the ‘ Con- 
naissance des Temps’ for 1849 and 1850) in relation to the cases in which 
the equations of motion of a particle or system of particles admit of integra- 
tion, are based upon Jacobi’s theory of the S function, that is, of the function 
which is the complete solution of a certain partial differential equation of 
the first order ; the equation is given, in the first instance, in rectangular 
coordinates, and the author transforms it by means of elliptic coordinates 
or otherwise, and he then inquires in what cases, that is, for what forms of 
the force function, the equation is one which admits of solution. A more 
particular account of these memoirs does not come within the plan of the 
present report. 
50. Desbove’s memoir of 1848 contains a demonstration of the two 
theorems given in Jacobi’s note of 1837, in the ‘ Comptes Rendus;’ and, as 
the title imports, there is an application of the theory to the problem of the 
planetary. perturbations; the author refers to the above-mentioned memoirs 
of Liouville as containing a solution of the partial differential equation on 
which the problem depends, and also to a memoir of his own relating to the 
problem of two centres, where the solution is also given; and from this 
he deduces the solution just referred to, and which is employed in the 
present memoir. Jacobi’s theorem gives at once the formule for the varia- 
tion of the arbitrary constants contained in the solution. The material 
thing is to determine the signification of these constants, which can of 
course be done by a comparison of the formule with the known formule of 
