96 
Proceedings of the Royal Society of Edinburgh. [Sess. 
described above, which is discontinuous, and repeats itself in a definite 
period, is a Fourier series. 
Write then 
t — n't + € 0 , 
R = A 0 + cos r + Bj sin t + . . . + A r cos rr + B r sin rr + ... . (15) 
The terms in R for which rj= 1 present no difficulty and are easily dealt 
with afterwards ; consider first, then, the special form 
E = A cos r + B sin r . . . . ( 1 5 a) 
together with 
x" + kx + n 2 x = R. (8) 
The frequency n is not at present supposed defined. The natural form of 
solution is 
x = a sin t — a 0 exp v . sin r, 
in which a contains no zeroes and r is a simply periodic function, the 
arrangement of zeroes of r in respect to t depending upon the definition of 
the former in terms of the latter. From the equations 
dv/dt 
cos r 
de/dt 
sin r 
Re v n 2 — n 2 ■ 
— k cos r h — - — sm r 
n a 
o 
n 
( 10 ) 
it is clear that if R=£0 there is no choice of n which makes e a purely 
periodic function so long as exp v is not purely periodic, that is, so long as 
the mean value of a is undergoing change ; for if at any moment we choose 
n' so that 
Up - v 71*2 7^2 
—j — sin r - k sin r cos r -\ — sin 2 r 
n a 0 n 
shall contain no constant term, a progressive character in exp ( — v) will 
afterwards disturb the balance. Hence so long as there is a progressive 
change in amplitude there will also be a progressive change in period. 
This, it will be observed, relates only to the case of maintained motion, 
with maintenance R containing a term in sin r. The same is clear by 
considering 
x + kx + n 2 x = R = B sin r + . . . = —x + ... , 
a 
where it is obvious that if a changes progressively, there is no fixed period 
to the motion. Hence the discussion divides into two parts : 
(1) Permanent periodic motion associated with a particular form of R. 
(2) Transition from one permanent motion to another when the 
coefficients of R are changed. 
Considering the first, the matter is now very simple. 
We suppose an asymptotic state to be reached in which any progressive 
