THEORY OF THE SEISMOGRAPH. 167 
that is, we must substitute for a, da/dt, and d’a/dt’, their values as given by the record, 
and we can then calculate d’&/dt?. If, as is generally the case, the record is not a simple 
regular curve, we must determine 
the values of a and those of its 
derivatives for points of the curve 
at very small intervals and then 
integrate the resulting values of 
@£/dt?, graphically or otherwise. 
This process is very long. If, on 
the other hand, the record is a 
simple harmonic curve, and it fre- 
quently approximates this for short 
times, we can integrate the equation 
directly. Equation (26) becomes, after pbeettiteie the values of the coefficients, 

Te 24 ta VE + Voge, + p'=0 (67) 
where we have put n? for gi/L, or (27/T,)?, by equation (32). 
Let us suppose first that there is no rotation, and the term Vga, disappears. Choos- 
ing the origin of time when the pendulum has its greatest elongation in the positive direc- 
tion, we can write 
a = dy) cos (2 7/P)t = a) cos pt (68) 
da/dt = — pay sin pt; da*/dt? = — p*a, cos pt (69) 
p' is a discontinuous function, having a constant numerical value, but suddenly changing 
sign with the velocity which it always opposes. We can represent it by the series 
: =<" (sin pt +4 sin 3pt+} sindpt+ + +) (70) 
where n’r, or (27/T',)’r, as in equation (48a), is the positive numerical value of p'; this 
series represents the broken line in figure 44 for all values of ¢. Substituting the above 
values in the equation of the pendulum, it reduces to 
2, 
yet Acos (pt — x) —2™" (sin pt +4 sin3pt+- - -) (71) 
T 
where 
A cos x = a(n? — p”); A siny = —2 xpd A? = a5 (n? — p’)? + (2 Kp)*} (72) 
Multiplying by dt and integrating from t=0 to t=t, we get 
dé dé A www 
Se n (pt — y) +— sin 
dt (> p aot eet p ae 
sae cos pt + = cos 3 pt - nic _ Anir 14141. sure (73) 
3? xp 

‘Integrating again, after replacing the last series by its value, 7/8, we get 
Vé—Vés—V (3) t= ~ ap A 


+ ae (sin pt +r —sindpt+- -.)— mir t (74) 
