1908-9.] 
On Lagrange’s Equations of Motion. 
337 
Now 
/ . , . aT dA, dA-. 
• (34). 
Also by (22) 
and therefore 
** + «$+ =0; 
dt dt 
-^r=-^ T+2 (cf), 
or 
But by (30) and (31) this gives 
d dT T , n X 
di^* 1 Ql 
d aX , T|V r\ '■ . . . . 
diWA* 2 =Q2 
. (35). 
• • ; 
Hence we have the very simple rule : modify the expression for T by 
substituting for s v s 2 , .... , their values from (27), and then proceed as 
if no velocities of co-ordinates s v s 2 , .... , had ever entered into the 
expression for the kinetic energy. The equations of motion obtained are 
of course applicable also to holonomous systems. 
18. To verify the results obtained, we write the first of (35) in the form 
ddT ar /, T ,_a x\_ n 
dt^~Wi~v l 
(36), 
and consider what it becomes when the system is made holonomous. 
We have 
= -(d 1 u + .... ) + %(c^j . . . (37); 
and since we have also now 
u= {a, - 2(eA)}g, + {a 9 - 2(eB)}ff 9 + . . . . 
v= {b 1 -^(fA)}q 1 + {b 2 -^(fB)jq 2 + 
we obtain 
0T 
d di 
= u 
+ V 
+ 
'a b 
dq 
. ( 38 ). 
22 
VOL. XXIX. 
