284 
ON THE DIFFERENTIAL EQUATIONS, &C. 
[43 
(The value of K in this section may I think be conveniently termed “the deter 
minant of the vis viva,” with respect to the variables x, y, z,.... It may be remarked 
that “ the determinant of the vis viva ” with respect to any other system of variables 
U, V, w, ... is = VK, V as before.) 
§ 6. Third form of the equations of motion. [Hamiltonian form.] 
Here writing 
dT ^ dT _ 
daf~* d^~ V ’ ’ 
and taking t, x, y,... y,... for the variables of the problem [and considering T to be 
expressed as a function of these variables: to denote this change it would have been 
proper to use instead of T a new letter H] the equations of motion reduce them 
selves to 
" d% _ dT „ dx _ dT 
dt dx + ’ dt dg ’ 
dy _ dT „ dy _ dT 
dt~~ddy^^’ dd~dy ; 
or putting for shortness 
they become 
dt : dx : dy : dz ... : d% : dy : d^ ... 
= 1 : E : H : to ... . X : Y : Z .... 
Hence writing the equation in M under the form 
where S = ^ + B 
we see immediately that (P, Q ... being as before independent of the velocities, and con 
sequently of y, ...), 
Hence 8M = 0, which is satisfied by M =1.