4 MODERN SEISMOLOGY 
that all seismographs are fundamentally the same, and if the 
frictional term could always be expressed as above no objec- 
tion could be taken to the statement. The different behaviour 
of instruments in actual practice is, however, mainly due to the 
fact that the frictional term is not of this simple form in all 
cases. 
The equation (B) is of well-known form, and full treatment 
may be found in any treatise on differential equations (e.g. 
Forsyth). 
The free motion is given by 
6+2 6+7O=0 
and the solution of this is of the form 
d=A e-“ sin {(z?-&)3(¢-y)} for n>e 
0=A e-@ sinh {(e? — 2”)3(¢- )} for ~<e 
@=A e-* G7) for z =e 
where A and » are arbitrary constants. 
The last case is of special importance in modern seismo- 
metry, and the instrument is then described as ‘‘dead-beat ” 
or “aperiodic”. 
In any case the quantities z and e are instrumental con- 
stants which may be determined experimentally by methods 
well recognized in ordinary laboratory practice. The quantity 
«x is in general a function of time and the recorded movement 
@ then consists of two parts: (1) depending on the special form 
of x, and (2) depending on the free movement of the instrument 
with constants depending on the initial conditions. The 
complete solution when + is any prescribed function is given 
by Rayleigh (“ Theory of Sound,” p. 74). 
We shall consider only the case in which + is a simple 
periodic function of the time, say + sin (/¢). 
The forced movement 6 corresponding to this is 
(= Py si aiaia) ; where tan 7 = 2 5s 
Le PP +460 PP 
The recorded movement thus differs in phase from the im- 
pressed movement. As the actual recorded quantity will be 
proportional to 6 say L@ 
the expression 
