334 
Proceedings of the Royal Society of Edinburgh. [Sess. 
same equations of motion. An example will be given at the end of the 
present paper. 
15. Let now the form of T be modified by the substitutions indicated 
above. Our object is to inquire what modification is required in the process 
of “ ignoration of co-ordinates ” by the non-integrability of the relations 
between the generalised co-ordinates and the functions of these co-ordinates 
and their velocities from which the kinetic energy is derived. We shall sup- 
pose, therefore, that the co-ordinates s v s 9 , . . . . , are absent from the kinetic 
energy, and from the function V of the co-ordinates from which the forces 
are derived, if that function exists. Writing them for a moment r 1 = dT/ds v 
r 2 — dT/ds 2 , . . . , we see that if the fundamental relations were integrable 
r v r 2 , , would be constants, since then we should have 
7 = 0 , 7 = 0 , .. 
0,q ds 2 
on the supposition that either 
7 = 0, 7|o, . . 
0s x ds 2 
or no generalised forces corresponding to s v s 2 , 
tions of motion are now, however, 
— —-X t T= o ' 1 
dt ds ± M 
d 8T r V n 
^aT, =X2T=0 
where 
J 
— 6-^u + yjr + . . • ■ | 
........ r 
. . , exist. The equa- 
• ( 21 ), 
• ( 22 ); 
so that x 2 > ■ • • • > are the operators for the s co-ordinates that 
<p v (p 2 , .... , are for the q co-ordinates. 
The conditions for constancy of the momenta dT/ds v 0T jds 2 , . ... , are 
therefore now 
\T = 0, X 2 T = 0, . . . (22'). 
These conditions are fulfilled by (22) when e v f v .... , are zero, which 
is the case in various problems of the motions of tops and gyrostats, where 
none of the coefficients e v f v .... , e 2 , f 2 , .... , contains the time or 
any of the co-ordinates q v q 2 , ... . We shall not assume, unless it is 
so stated, that e v f v .... , e 2 , f 2 ,...., are absolute constants. 
16. We shall assume, however, that 
0T 0T 
57"" c i ’ 7T “ c 2 > 
l 
2 
(23) ; 
