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ON THEORETICAL DYNAMICS. 27 
elliptic motion; the author is thus led to a system of canonical elements 
similar to, but not identical with, those obtained by Jacobi. 
51. Serret’s two notes of 1848 in the ‘ Comptes Rendus.—These relate 
to the theory of Jacobi’s S function, that is, of the function considered as the 
complete solution of a given partial differential equation of the first order. 
In the first of the two notes, which relates to a single particle, the author 
gives a demonstration founded on a particular choice of variables, viz., 
those which determine orthotomic surfaces to the curve described by the 
moving point. The process appears somewhat artificial. 
52. Sturm’s note of 1848, in the ‘Comptes Rendus,’ relates to the theory 
of Jacobi’s S function, that is, of this function considered as the complete 
solution of a given partial differential equation of the first order. The 
force function is considered as involving the time ¢, which, however, is no 
more than had been previously done by Jacobi. 
53. Ostrogradsky’s note of 1848.—This contains an important step in the 
theory of the forms of the equations of motion, viz..it is shown how, in the 
case where the force function contains the time, the equations of motion 
may be transformed from the form of Lagrange to that of Sir W. R. Ha- 
milton. If, as before, the force function (taken with the contrary sign to 
that of Lagrange) is represented by U, then putting, as before, T+U=Z 
(the author writes V instead of Z), in the case under consideration Z will 
contain not only terms of the second order and terms of the order zero in the 
differential coefficients of the coordinates g,..., but also terms of the first 
order, that is, Z will be of the form Z=Z,+Z,+ Zp, and putting H=Z,—Z,, 
this new function H being expressed as a function of the coordinates g,.. 
and of the new variables p,..., then the equations of motion take the Ha- 
miltonian form, viz.— 
dqg_dH dp_ dH 
dt dp’dt™~ = dq° 
In the theory of the transformation, as originally given by Sir W. R. Hamil- 
ton, Z,=T, Z,=0, Z)=U, and, consequently, H=Z,—Z,=T—U as 
before. 
54. Brassinne’s memoir of 1851.—The author reproduces for the La- 
grangian equations of motion 
ddZ dZ __ 
dt di! di’ 
the demonstration of the theorem 
d (.dZ dZ., 
af? Gphb—AGabE+...)=05 
and he shows that a similar theorem exists with regard to the system 
@ dZiddz dZ_ 
dé dé"' dt di” dé 
and with respect to the corresponding system of the mth order. The 
system in question, which is, in fact, the general form of the system of 
equations arising from a problem in the calculus of variations, had pre- 
viously been treated of by Jacobi, but the theorem is probably new. In 
conclusion, the author shows in a very elegant manner the interdependence 
