43] WHICH OCCUR IN DYNAMICAL PROBLEMS, 
the equations which determine X, Y, Z ... are 
283 
aX + hY+gZ... A 8L -~-P = 0, 
hX + bY + fZ...+8M-~-Q=0, 
gX +fY+cZ ...+8X-^-R= 0. 
Hence, differentiating with respect to x', 
dX 
Ah 
dY 
dZ 
8a 
dx' 
dx' +3 
dx' ’ 
.. + 
dX 
dY 
dZ 
8h + 
dM 
dL 
dx' 
A b 
dx' + S 
dx' 
. + 
dx 
dy 
dX 
+ / 
dY 
dZ 
8gA 
dX 
dL 
dx' 
dx' + c 
dx' ’ 
.. + 
dx 
dz 
= 0, 
or representing by K the determinant formed with a, h, g, ... h, b, f ... g, f c, ... and 
by A, H, G, ... H, B, F, ... G, F, G ... the inverse system of coefficients, we have 
K ~ + A8a + H8h + G8g ... + 
and similarly 
dY 
A H -^) + g(~-^) ... =0, 
\dx dy ) \dx dz; 
K l ^, + HSh + BSi + FB/...+H(~-~)+ * +f(~ ~~) ... = 0, 
dy dx J 
V dy dz ) 
K s + Gtg + Fèf+ C& ... + 0 (§ - f) + <f-f) + 
0. 
Hence, adding, 
K (^ + gf' +( ^'---) + A8a+B8b + C8c... +2F8f+2G8g + 2H8h ... = 0 ; 
and thus we have as before, though with symbols bearing an entirely different signification, 
jr (dX dY dZ Y . 
K [d3 + w + dz' + -) + * K =°> 
and thence K8M — M8K = 0, and M = K. 
36—2