282 
ON THE DIFFERENTIAL EQUATIONS 
[43 
Thus the equation in M reduces itself to 
KBM - MBK = 0, 
which is satisfied by M = K. It may be remarked that K reduces itself to the sum 
of the squares of the different functional determinants formed with the differential 
coefficients of ©, ... with respect to the different combinations of as many variables 
out of the series x, y ... . 
§ 5. Lagrange’s second form. 
Here the equations of motion are assumed to be 
d dT dT p 
dt dx' dx ’ 
ddT 
dt dy' 
dT 
dy 
-Q = 0, 
d dT dT 
dt dz' dz 
where 2T represents the vis viva of the system, x, y, z, ... are the independent variables 
on which the solution of the problem depends, and x', y', z',... their differential coeffi 
cients with respect to the time. It is assumed as before P, Q, B ... do not contain 
/ / / 
X, y, z, ... . 
Suppose these equations give 
dt : dx : dy : dz ... : dx' : dy' : dz' ... 
= 1 : x' : y' : z' ... : X : Y . Z 
then the equation which determines the multiplier M takes as before the form 
BM+M 
fdX dY dZ 
\dx' dy' dz' 
= 0. 
To reduce this equation, substituting for T its value which is of the form 
T = \ (ax' 2 + by' 2 + cz' 2 ... + 2fy'z' + 2gz'x' + 2lix’y' ...), 
and putting for shortness 
L = ax' + liy' + gz' ... , 
M = hx' + by' + fz ... , 
N = gx' +fy' +cz' ...