332 Proceedings of the Royal Society of Edinburgh. [Sess. 
motion of the system is referred to fixed axes, and the “ velocities ” 
u, v, ... . chosen are such as correspond to applied forces U, V, . . . . 
really applied. For instance, it is always correct to express T in the form 
|-2{m(^ 2 + ^ 2 -f z 2 )}, and in the case of a rigid body to insert for x, y , z 
their values in terms of the motion of the centroid, and of the angular 
velocities of the body about the chosen axes, taking care that the proper 
applied forces, whether derived from the potential energy or otherwise 
assigned, are employed. 
11. For the sake of what follows, we compare equations (13) with the 
usual form a little in detail. We have here 
da, . , da 9 . , 
a i = ^ c h + ^ ( l2 + ■ 
dep dq 2 
i db, . db 2 . 
dq l dq 0 
+ 
da Y 
0s x 
S-, + 
0a x 
° 2 
A + 
+ 
d hs h3 S + 
ds 1 1 3'.9 2 2 
Also 
or, by (11), 
0T . du . dv 
k ^ v — + 
dq Y dq 1 dq x 
0T 
% 
= u 
da, . „ da 0 . 
v- 1 9i + 2 q 2 + 
0^1 dq l 
. /06j . 0& 2 • , 
+ v ( ^ <h + V- 1 ?2 + 
%i dq } 
de 1 
+ i & 
d( h 
de „ 
+ ^ s-, + s 2 -f 
dq i 
+ — • ^2 + 
dq Y 
+ 
Thus, by (14) and (15), we get for the first of (13) 
± d l _ E _ * J J d ri _ ^ + a A - d rA 
y^0T 
dt dp 
dq x 
+ sp 1 
. j . fdb 1 0$, , . 
~ v \ qi wr ^ ] )+?* 
dq l 
da 1 de ^ 
0S] dq 1 
db x 
-3 
+ i ( da * 
de. 
db 2 \ . (db T 
dq 2 dq-) ^ 3 \dq 2 
+ s. 
djh 
ds 1 
Vi 
d <h 
+ s. 
0^1 
dip 
dq x 
d\_df 2 
ds 2 d gi 
+ 
+ 
( 14 ). 
(15). 
(16). 
The other equations of (13) may of course be written down in a similar 
form by symmetry. 
If the conditions 
da 1 da , 
da, da, 
