335 
1908-9.] On Lagrange’s Equations of Motion. 
where c v c 2 , ... . are constants. These conditions are fulfilled in a large 
number of problems regarding rotating fly-wheels in which the co-ordinates 
q v q 2 , . . . . determining the positions of the axes of rotation have no 
influence on the momenta 0T lds v .... When the system is holonomous 
the constancy of these momenta is secured by the fact that the differential 
equations (22') become dT/ds 1 = 0, .... Equations (23) extended are 
+ .... )q l + +/i ^2 "t • • • • )$2 • • • ■ + ( e i“ "h/p + • • • • )$\ 
+ (<Sp 2 *L/"l f 2 "*“•••• )^2 • • • • ~ C l 
( e 2 a i + • • • • ) 4 l + ( e 2 a 2 + • ' • • ) ( h + • • • ‘ + { e \ e 2 +/1/2 + • • • • ) 6 \ ' 
+ ( e 2 2 +/2 2 + • • • • )h + • • • ' = H 
(24). 
• y 
The coefficients of q v q 2 , ... , s v s 2 , .... , are those of the products 
q x s v q 2 s v . ... , s x s 1; s ± s 2 , . . . . , in the first line, and of q ± s 2 , q 2 s 2 , . ... , 
s 2 s 2 , . . . . , in the second. Denoting these coefficients as in the scheme 
(* 2 i> 6 i)> (^2’ ^2)’ • • • > ( s v s i )’ ( s v %)> • • • • 
Lu ^ 2)5 (^2’ ^ 2 )’ • • • j ( S l> ^ 2 )’ ( s 2> ^ 2 )’ ' • * • 
we can write equations (24) as follows : — 
(VSi)«i + («2» S l) 6 '2+ • • • • = C l~(<ll> S l)4l-( ?2» S l)?2~ • • • •] 
(• s l> ^ 2)^1 ( c<? 2> S 2)% • ' • • = C 2 ~ Ll> ^' 2)^1 ~ (?2» ^ 2)^2 * (25). 
From these s p s 2 , . . . . , can be determined in terms of c v c 2 , . . . . , and 
q v q 2 , ... . These values then substituted in the expression for T convert 
it into a function of q v q 2 , ... , c v c 2 , . . . , so that all trace of the variables 
s v s 2 , .... is now removed. We have to inquire what form the equations 
of motion take when this substitution is made. First we form expressions 
for s v s 2 , ... . Let (c v c x ), (c v c 2 ), . ... , denote the ratios of the consecutive 
first minors of the determinant of equations (25) to that determinant, 
and put 
Then 
K = i{(c v 
. 9K 
s, = — 
1 0Cj 
. _aK 
dc 2 
C l) C l 2 + 2( c i) C 2) c l c 2 Jr • • • •} 
~ (4lM + ^2^1 + . ... ) 
~ (^1^2 ^2^2 +••••) 
(26). 
(27); 
^i = ( c n G i)( < lv s i) "t (^i, c 2 )(^ f 1 , s 2 ) + . . . , A 2 = (c 2 , c i)({Zi5 s i) N 
+ (r 2 , r 2 )(^i, s 2 ) 4- .... 
1> 1 = (c p Cj)(^ 2 , s x ) + (c-p r 2 )(g , 2 , s 2 ) + . . . , B 2 = (c 2 , c i)(^ 2 , Sj) > • 
+ (C 2 , C 2 )fe» S 2 ) + • • • • 
J 
( 28 ). 
where 
