98 
Proceedings of the Royal Society of Edinburgh. [Sess. 
excursion of x ; in this case the amplitude a of the oscillation is zero and 
the period is altogether deranged. 
(3) For A, B, both positive, when the maximum of the maintenance 
precedes the zero of x, n <n, i.e. the maintained motion loses on the natural 
period ; and generally 
(4) As — B/A increases, the maintained motion gains. 
(5) A variation of A or k will affect the amplitude a 0 and thence the 
period also, through the circular error ; but this may be treated separately. 
(6) To a degree of approximation that is almost complete, the whole 
permanent motion depends upon the first two coefficients A, B of the 
Fourier series which expresses the maintenance. 
Now consider No. (2) of p. 96, viz. the transitional motion that takes 
place before a permanent motion has established itself under a specified 
maintenance. The chief interest of this is the transition from one motion 
of permanent character to another when the frictional coefficient k or the 
coefficients of the maintenance A, B undergo an alteration. We know 
from the pages above how much the period of the motion will finally 
change, but the further question arises, — How much does the epoch change 
in the interval of transition ? 
Let A, B, k change to A + AA, B-f AB, k + Ak', retain only the non- 
fluctuating terms; then from (10), p. 91, 
whence 
d( Av) jA + AA^ 
dt 2 ria 0 
1(k + A/c), 
Ce~!( K + A '0 i , 
where C is an arbitrary constant ; using the condition, 
Av = 0 for A A = A/c = 0 when t — 0, 
we have 
( 1+ T>“d 1+ x) + (t K -x>- i(K+i ' t,( • 
showing the way in which a 0 = A/uk changes into 
a = a Q e Ay — (A + AA)/A(k + A/c). 
The final change in amplitude is 
A a A A A/c 
a A k 
In the same way 
AB A A A/c /A/c AA 
7b a~ + T 
d Ae , B 
-dt = ^ K A 
if we omit squares, etc., of AA/A, AB/B, A k/k. 
. (17a) 
