170 Proceedings of the Boyal Society of Edinburgh. [Sess. 
so that 
S + S' = 2(^)-2( Mo ) (4) 
We have also, of course, 
t 
S - S' = j {^(pq -pq) - 2E} dt ; .... (4') 
o 
but this can hardly be regarded as differing from (4). If we integrate 
— H(pq)dt by parts we get Hpq + \H(pq)dt, so that (5) becomes (4). 
Besides the relation already known dS jdt= — dS'/dt, we obtain from (4) 
the equation (taking now S, S' as the “ indefinite integrals ”) 
0S_ _0S' = & 
da~ daj J ’ 
( 5 ) 
where 6 is a constant. For, having regard to the variables in terms of 
which S and S' are expressed according to their definitions, we have by (4) 
It is here assumed, however, that the constant a 5 is not a momentum 
which is constant in consequence of the non-appearance of the correspond- 
ing co-ordinate in H. If % be such a momentum a, and the co-ordinate be 
q jt we shall have 
Thus while dS/da is a constant, dS'/da is not : an example is given below. 
I have not seen this theorem before, probably because S' is little used. A 
more complete account of these constants is deferred.* 
In the determination of S or S' in actual cases I prefer, as a rule, to 
employ (16) of §1, or (2) of the present article, rather than to set up the 
related differential equations for oS jdq, or dS'/dp, and solve them, which is 
the usual practice. Of course the two processes are fundamentally the 
same, but the calculation of the time-integrals, in the equations referred to, 
is the more intelligible, at least to the student. It may be noticed that 
(16), § 1, gives at once, as a matter of course, when certain co-ordinates 
* Note added April 18. — Reciprocal dynamical theorems are easily derived by means of 
the two functions. Thus, let the constants in S, S' be the as (the initial co-ordinates). 
Then, for the initial momentum 6 *, we have 6* = - dS/dai = dS'/dai, and therefore 
dbi/dpj[=d(dS' /dpj)/dpj] = dq j /da i : also db it ldqj[= - d(dS/dq j )/da i ] = - dpj/da^ If the constants 
in both functions be the 6s (the initial momenta), two other theorems are obtained in a 
similar way, namely, dajdpj= -dqjdbi ; dai/dqj^dpi/dbi. 
I find, however, that this mode of deriving (two of) these relations has been anticipated 
by v. Helmholtz ( Grelle , 1886), who also gave physical interpretations. Applications are 
given by Lamb ( Lond . Math. Soc., 1888). 
