12 REPORT—1857. 
which, if they could be integrated, would give S as a function of a, y, z, 
a, b,c, t, and thus determine the motion of the system. 
24. Hamilton’s form of the equations of motion.—This is in fact the 
form obtained by carrying out the idea of introducing into the differential 
equations, in the place of the differential coefficients of the coordinates, the 
derived functions (with respect to these differential coefficients) of the 
vis viva function T. Taking n to denote any one of the series of coordi- 
nates, then the original system may be denoted by 
daY_daT_dv 
dt dni dn dn’ 
(U is the force function taken with a contrary sign to that of Lagrange), 
and writing in like manner @ to denote any one of the new variables con- 
nected with the coordinates 7 by the equations 
dT 
anit 
then T, in its original form, is a function of n,...7!,..., homogeneous 
of the second order as regards the differential coefficients 7!..; and, con- 
sequently, these being linear functions (without constant terms) of the new 
variables w, the vis vive function T can be expressed as a function of n,..+ 
@,..., homogeneous of the second order as regards the variables a,... 
And when T has been thus expressed, the equations of motion take the form 
dn_ dH da dT dU 
dt da? dé" dnt dy’ 
which is the required transformation. The force function U is independent 
of the differential coefficients n',.. and, consequently, of the variables a, .., 
hence, writing H=T—U, the equations take the form 
dn_ dH dw_ dH, 
dt da dt dn 
which correspond to the condensed form obtained by writing T—V=R in 
Lagrange’s equations. It is hardly necessary to remark that H is to be 
considered as a given function of 7,...@,..+ viz. it is what T—U be- 
comes when the differential coefficients 7',... are replaced by their values 
in terms of the new variables w,... 
25. I have, for greater simplicity, explained the theory of the functions V 
and S in reference to a very special form of the equations of motion; but 
the theory is, in fact, applicable to any form whatever of these equations ; 
and, as regards the function V, is in the first memoir examined in detail 
with reference to Lagrange’s general form of the equations of motion. The 
function S is considered at the end of the memoir in reference only to the 
special form. The new form of the equations of motion is first established 
in the second memoir, and the theory of the functions V and S is there con- 
sidered in reference to this form. The author considers also another 
function Q, which, when the matter is looked at from a somewhat more 
general point of view, is not really distinct from the function S. 
* | find it stated in a note to M. Houel’s ‘ Thése sur l’intégration des équations différen- 
tielles de la Mécanique,’ Paris, 1855, that this form of the equations of motion had been 
previously employed in an unpublished memoir by Cauchy,,written in 1831. 
