14 REPORT—1857. 
general formulz) both sets of variables, and the consequence is that, as the 
author remarks, the varying elements employed by him are essentially different 
from those made use of in the theories of Lagrange and Poisson; the ve- 
locities cannot, in his theory, be obtained by differentiating the coordinates 
as if the elements were constant. The investigation applies to the case 
where the attracting force is any function whatever of the distance, and the 
six elements ultimately adopted form a canonical system. 
28. The precise relation of Sir W. R. Hamilton’s form of the equations 
of motion to that of Lagrange’s, is best seen by considering Lagrange’s 
equations, not as a system of differential equations of the second order 
between the coordinates and the time ¢, but as a system of twice as many 
differential equations of the first order between the coordinates, their 
differential coefficients treated as a new system of variables, and the time. 
It will be convenient to write —U, instead of Lagrange’s force-function V, 
and (to conform to the usage of later writers who have treated the subject 
in the most general manner) to represent the coordinates by q,..., their 
differential coefficients by g', ..., and the new variables which enter into the 
Hamiltonian form by p,...; then the Lagrangian system will be 
dq, & dV, dT _du. 
dt!’ dtdq dq dq’ 
or putting T+ U=Z (this is the same as Lagrange’s substitution, T—-V=R), 
the system becomes 
dq, a de. oz 
a’ hag ~ a 
while the Hamiltonian system is 
dq_ dl dp aT, dU. 
dt~ dp’ dt dq‘ dq’ 
or putting as before T—U=H, the system is 
dq_dH dp_ dH. 
ft help EOS ag 
where, in the Lagrangian systems, T and U, and consequently Z, are given 
functions of a certain form of ¢,g,..g',.., and in like manner, in the Hamil- 
tonian system, T and U, and consequently H, are given functions of a cer- 
tain form of ¢,g,..p... The generalisation has since been made (it is not 
easy to say precisely when first made) of considering Z as standing for any 
function whatever of ¢,q,...q',.., and in like manner of considering H as 
standing for any function whatever of ¢,g,..p,... It is to be noticed that 
in Sir W. R. Hamilton’s memoir, the demonstration which is given of the 
transformation from Lagrange’s equations to the newform depends essentially 
on the special form of the function T as a homogeneous function of the 
second order in regard to the differential coefficients of the coordinates ; 
indeed the transformation itself, as regards the actual value of the new fune- 
tion T (=T expressed in terms of the new variables), which enters into the 
are referred. The method of Sir W. R. Hamilton is made use of in M, Houel’s ‘ Thése 
d’Astronomie ; Application de la Méthode de M. Hamilton au Calcul des Perturbations de 
Jupiter,’—Paris, 1855. 
