336 Proceedings of the Poyal Society of Edinburgh. [Sess. 
17. Let now T' denote the expression for the kinetic energy when 
s v s 2 , ... . have been expressed as in (27), and T denote the former 
expression. Since q v q 2 , .... , appear in T' exactly as they did before 
the substitutions, and now appear besides in consequence of the substitu- 
tions, we have 
0T 0T 0*, 
— = — + c-,-1 + c 
d 9l 0£ 0^ 
where for s v s 2 , ... . 
inserted. Thus 
0T 0S, 
— + c l. 1 
+ c*! + 
(29); 
a i+ d JL 
2 dq 2 ‘ dq 2 dq 2 ' ^ ' ~*dq 2 
, their values from (27) are supposed to have been 
d 0T = d_ 0T + d I \ = _ ^(dA\ 
dt dq l dt dq Y dt\ \ dqj / dt dq l \ dt / 
by (27), so that 
— 0T- + ^c— V — — ^ 
dt dq Y \ dt ) dt dq^ 
_£ 0T' 
dt dq 2 \ dt) dt dq 9 
j 
Now from the original expression for T we have 
£ ^ 
dt 0g 1 
do T 
dt dq 2 
- <^T - Q 1 
- <f > 2 T = Q 2 
J 
(30). 
(31). 
But if we substitute in d\ T, for example, the values of s v s 2 , .... , 
from (27) we obtain after reduction 
_< AiT= - [«i{«i -2(eA)} + &i{fej - 2(/A)} + 
“ [ffiK - 2 ( eB ) } + ^l{ & 2 - S(/B) }+•■•• fe 
-[ fc (32). 
If T' be subjected to the same operation <fi v with the values of the 
coefficients of q v q 2 , .... , q k as they exist after the substitution for 
s v s 2 , .... , we get 
- ^>i r r =-—{«!- - 2(eA)}^ + {« 2 - ^(eB)}^ 2 + . . . . ] 
-^{&i-S(/A)}[{6 1 -2aA)}tf 1 + {6 1 -S(^B)}*+ ] 
(33). 
Collecting on the right of this expression the coefficients of q v q 2 , .... , 
we see by (32) that 
-*r~-*T + ^ + A*)* + S 
f dA + A df 
dt dt 
\v + . . 
