331 
1908 - 9 .] On Lagrange’s Equations of Motion. 
The parameters are divided in (9) into two sets — the q s and the s s — 
for a purpose which will appear later in connection with ignoration of 
co-ordinates. By (9), 
u = a 1 q l + a 2 q 2 + ....+ + e l s x + e 2 s 2 +....+ qs,- 
v = b 1 q 1 + b 2 q 2 + .... + b i q i +f l s 1 +f 2 s 2 + .... +fjSj - . 
( 11 ), 
if we suppose, as we do for the present, that t does not appear in the 
kinematical equations. 
9. Also we have, by the signification of it, v, ... , equations of motion 
ii = U, v = Y , 
and therefore obtain a series of equations of the form 
ciqil + bqv + ....= rq U + bfi + . . . . = 
ctqii -}- bqv -f- . . . . — ct^XJ -1- bi\ r . . . . = Q 2 _ 
(12), 
where Q p Q 2 , . . . , are generalised forces according to the meanings of 
u, v, ... . These are i+j such equations, since there are now supposed 
to be i+j independent parameters q v q 2 , ... , s v s 2 , ... . 
By (10) and (11) we have 
d 0T , . . 4 . v n ] 
' ' ' ' )_Ql 
d 0T / . . 9 . \ _ n " • • • (13). 
— — — (a 2 u + b 9 v +•...) — Qo 
at dq 2 
' ■ , 
These are equivalent to the equations which, as Ferrers showed, must be 
substituted for the ordinary Lagrangian equations as typified by (6). 
They are applicable in all cases, whether the system is holonomous or not, 
provided always that the co-ordinates chosen are capable of expressing the 
configuration of the system, or position of the body at any instant. 
10. It must, however, be remembered that different modes of breaking 
up the kinetic energy into a sum of squares according to (10) are in 
general not equivalent, but involve different sets of forces. For example, 
the term JA(0 2 + sin 2 0 . f 2 ), which occurs in the expression for the kinetic 
energy of a gyrostatic pendulum, or of any kinetically symmetrical body 
rotating with one point fixed, and which is already a sum of two squares, 
may also be written in the form JA (6 cos (p + f sin 0 sin ^>) 2 + JA(dsin f 
— <p sin 0 cos (p) 2 . The forces involved in the two cases are different, and 
hence, unless the appropriate corrections necessary on this account are 
introduced, the distribution of the kinetic energy into a series of squares 
is not arbitrary. Correct results may, however, always be obtained if the 
