280 
ON THE DIFFERENTIAL EQUATIONS 
[43 
where 
v ^ d% d<& 
X = p + X — H- yL6 
dx ax 
tt ri dS d<$? 
3 ~ Q + ^'dy +/J '~dy f 
where 0 = 0, <É> = 0, ... are the equations of condition connecting the variables, and 
X, /a, ... coefficients to be determined by substituting the values of &c. in the 
equations = 0, = 0, &c. It is supposed that as well P, Q, ... as 0, are inde 
pendent of the velocities. 
In order to reduce these to an analogous form to that previously employed, we 
have only to write 
which gives 
dx , dy . 
Tt= x ' tt= y 
dt : dx : dy : dz ... : dx' : dy' : dz' ... 
= 1 : x' : y' : z' ... : X : Y . Z ... 
Supposing that M is independent of x', y', z\ ... the equation on which it depends 
becomes immediately 
?71/r , nrfdX dY dZ \ 
SM + M {d7 + d? + 3P + -) = 0 - 
where for shortness 
rs / d 
S ~dt + X dx + y Ty + Z S+-" 
To reduce this we must first determine the values of X, y, ... , and for this we have 
d-0 _ dM d?x d© cZ 2 ?/ 
dt 2 ckr di 2 dy dt 2 ^ i. 
= 0, &c. 
i. e. 
T) /-> <^0 
* 0 + P £ + < ^ + - 
,. -P «X -P A/a -p //a* -p • 
.. =0, 
** + P a« + «f + " 
, . -p h\ + 6/A -p fv -p .. 
. =0, 
W + P f + «f+- 
■ • -b = ft L Y cv -P .. 
. = 0.