88 H. A. WILSON 
shows that making »/w large will diminish the maximum values of y very 
greatly. For example if u/w=10 we get 
y = (A/99)(cos 10wt — cos wt) 
so that the maximum amplitude is only about one percent of that of the 
waves in the ground. In this case we get short oscillations due to the term 
cos 10wt superposed on the curve given by —(A/99) cos wt. Hence a smooth 
curve drawn across the small oscillations would give —(A/99) cos wt which 
resembles the curve given by x = A(1—cosw#) although displaced downwards 
and with only one percent of the amplitude. This suggests that the results 
obtained with high frequency undamped seismographs can be improved by 
drawing a smooth curve through the small oscillations of frequency equal to 
that of the seismograph. 
Fig. 6 shows the curves given by 
x = 1 — cos (30#)° 
and 
— y = (%)(cos (90#)° — cos (30#)°) 
which are those for the case A =1, w=27/12 and u» = 27/4 in which the period 
of the seismograph is one third that of the waves. 

Fig. 6. Fig. 7. 
Now consider the case of a critically damped seismograph. In this case if 
w=21/12 so that x = 1—cos(30#)° and k= p= 27/36 so that the seismograph 
period is three times that of the waves then the equation for y gives 
” cos (30¢ -+ 36-87)° + ee geen 
= — cos . er t18, 
es 10 100 
Fig. 7 shows the curves given by these equations. Comparing Fig. 7 with 
Fig. 3 we see that the critical damping cuts out the oscillations of the seis- 
mograph, set up at the start, so that we do not get interference between two 
y 
Fig. 8. 
vibrations of different frequencies. The damping however diminishes the 
interval at the start in which x and —y agree. Thus without damping in Fig. 
3 the two curves agree from ¢=0 to t=4 but with critical damping in Fig. 
232 
