338 Proceedings of the Royal Society of Edinburgh. [Sess. 
If now dajdq l — dajdq 2 , .... db 2 /dq 1 = db l /dq 2 , .... , (37) becomes 
(0 0 
u.ii + b-,v + . ... —u\ ffi- — 2(eA) -I- q*- — S(6l>) + . 
{ dq 1 dq 1 
■-! 
by (22')- Hence finally 
d 8T 0T' . / /0A 0B\ 
dt dq Y dq 1 ( \dq 2 dqj 
Qi 
(39) 
(40); 
and, similarly, for the other equations. The terms in q v q 2 , . . . . are called 
gyrostatic terms. The coefficient of q x in the first equation, of q 2 in the 
second, and so on, is zero. 
This simple modification of the expression for the kinetic energy by 
which the equations of motion are obtained when certain co-ordinates are 
ignored, may be compared with the modification given by Routh for the 
case of holonomous systems {Stability of Motion, p. 60). If T and T r have 
the meanings assigned above, we have now 
and obtain 
0T_0T 2 / ds_ 
d dT 0T # 
dt 
£3T_0T = Q | 
dcjj dq 1 1 \dt dq Y dqj i dt cq 1 dq x |> 
(41). 
But, as has been shown above, § 17, 
0s \ 
= — *Z\ C I . Z.I c 
dt dqf 
d ds\ dA\ 
2( ^^=_a (c-), 
(42), 
so that (41) becomes 
A - A - ^ + 4,2 < c(A _ A) l + 4 5 1 ,(A _ A) 
dt dq x dq Y I \0g 1 dqj ) ^ | \dq 2 dqj 
+ 
-Qi| 
(43). 
Equations (41) show that if we modify the expression T to T' by 
substituting in it the values of s v s 2 , ... . given by (27) and then write 
rnV rrv 
1 JL ^1^1 ^0^2 . * • • 
we can use T " to obtain the equations of motion for the co-ordinates 
q v q 2 , .... , q k for a holonomous system by the ordinary process. Equa- 
tions (43) show that the so-called gyrostatic terms flow from the added 
expression — c 1 s 1 — c 2 s 2 — . In equations (35) these added terms are 
dispensed with, and the equations are applicable to all kinds of systems. 
