100 Proceedings of the Royal Society of Edinburgh. [Sess. 
or if we write 
they are 
here 
£ = e Ki E, r] = e Kt F, ( = e Kt G, 
£ + k'C = He Kt = A, 
r t '-2n£ = <f>e* t = M 
k<~ 4- 2 nrj + = A ■ 
(186) 
A(£-£) = M rj, or A v = W(£ + £). 
From these the natural period for the case 
A = M = 0 
is immediately deduced ; viz. putting 
i, p, t ex P (ipt), 
we have 
0 = 
or 
ip 
* 
K 
* 
ip 
— 2 n 
K 
2 n 
ip 
p 2 = 
= 4?i 2 
- K 2 . 
= ip( —p 2 + in 2 - k 2 ), 
We may write 
rj = £ sin 2<f>, £ = £cos2</>, 
and putting 
(f) - nt + e, 
take £, e as variables, when the equations will run 
2e = k sin 2cj) — M/£ 
Z I 
The last may be written 
and 
and 
£/£= -K COS 2<f> + A/£ . 
E'/E = - 2k cos 2 </> + H/E 
A = = R.c'e'y M = i?e Kt = Hxe Kt 
H sin cj> = <& cos </>. 
(18c) 
These may be compared with equations (10), p. 91, when the agreement 
is at once seen. 
H is the rate of supply of energy by the maintenance, and <l> the virial 
of the maintaining forces. 
As final form of the equations we may take 
E'/E = —2k cos 2 <f> + H/E, 
2e' = k sin 2c j) — <h/E, 
giving the variations of energy and epoch. In these expressions the 
variations that take place within the period of <fi, that is, within 2 secs, 
are, generally speaking, immaterial ; in this case the equations may be 
