174 Proceedings of the Royal Society of Edinburgh. [Sess. 
But by (15) 
r 2 # = ^ = 
m 
= \ /K 2 - a . , and a = mr 2 sin 2 # . </>. 
V sm 2 # 
Hence 
0S 
0^ 
= cf) - J cj>dt — 0, 
if (p, as it is here taken, is zero when t = 0. This gives 
0S' , 0S , 
Ta = *~T« =4 ‘’ 
(27) 
(28) 
tlie value of the co-ordinate associated with a. 
This may be verified by differentiation of S' with respect to a, as follows. 
From (23) we get 
0S'_ I 
dp 2 
0T J 
Jm 2 li 2 - j 
since m 2 K 2 — y> 2 2 = a 2 / sin 2 #. 
But we 
p 2 dp 2 = - 
so that 
tan# j 
dp 2 = 
adO 
a 
p 2 sin 2 # 
= I d lh x 
; 2 - a 2 J a 
dp 2 tan # 
(29) 
dO 
Thus we get, as in (28), 
0S' 
0a 
sin 2 # tan 0 ’ 
mr 2 sin 2 # . p . dO _ 
mr 2 sin 2 # . # 
pdt, 
(30) 
reckoning: from the same zero as before. 
Finally we notice that (9) leads directly to the modified Lagrangian 
function L 1} to be used when certain co-ordinates (q lt q 2 , .... , q g , above), 
which are represented in H by their velocities only, are ignored. We have 
Li = L - atf-L - a 2 q 2 - - a g q g (31) 
where L denotes the unmodified function. For if we write 
^1 = ® — a i^i ~ a 2^2 — ~ a g9.g-> • (^2) 
any q of suffix higher than g enters only in S 1? so that S x is the principal 
function for the non-ignored co-ordinates. It is understood, of course, that 
wherever any of the velocities q v q 2 , . . . . , q g occur in they are to be 
replaced by their values in terms of q g+1 , q g+2 , . . . . , q k , q g+l , q g % . . . . , q k , 
given by the equations 
0T 
0T 
dq, ” dq 2 * 
derived from T as expressed in (2), § 1. 
0T 
d 4g 
(33) 
( Issued separately May 17, 1912.) 
