THEORY OF THE SEISMOGRAPH. 165 
From this we deduce as before 






EOE gy ates (oe Ade 
Solving equation (50) for 27, we get 
Op fats As eAy ete. 
1l+e 1l+e 
ss wy 
= a = = ai ey =--. ete, 
adding numerators and denominators 
Ay CAs.) at TA CAs 
~ G+9d—-/d—9) «f1 e=1 
replacing value of ¢" from equation (54) we get 
ee EBay 51) 
vee +1 (A, = 43) —-Cin — Aon+1) 
Equations (55), (56) and (57) are perfectly general; and n may be given any integral 
value greater than 0. The factor (e—1) in equation (57) reduces the accuracy in the 
determination of r when e¢ is nearly equal to 1; but e can be determined with considerable 
accuracy from equation (54) if we have a good record of free vibrations without outside 
disturbance. r being thus determined, we can find p’ and ¢ from equation (48a). Thus 
the damping and frictional constants can be determined from the measure of 3 ranges. 
Returning now to equation (35), let us consider the case where the friction is so great 
that the movement is no longer periodic so that we can not determine « and p’ by the 
above methods. We shall then have « >27/T,, and the solution of the equation (35) 
under this condition is 
a= A,e~™ + A,e~™ (58) 
where 
Mm =K+tEVR—n, m= K-VK —V? (59) 
and n is written for 27/T,; A, and A, are arbitrary constants whose values are to be 
determined to correspond to the special conditions imposed. Neglecting solid friction 
for the present, we can determine the value of « by displacing the pendulum an amount 
a, and then setting it free; that is, at time t, we have a=a, and da/dt=0. If we deter- 
mine A, and A, to satisfy these conditions, equation (58) becomes 
a= 

—mat __ —mt 
Ay cae (mye mye—™*) (60) 
This represents the difference of two exponential curves, and since m, is greater than m,, 
the second term in the parenthesis is always smaller than the first and a is always posi- 
tive; and therefore the pendulum remains on the positive side of the position of equi- 
librium, gradually approaching it, but only reaching it when ¢ is ©. 
To determine « we must first determine n or 27/T,. To do this, reduce the value of 
« sufficiently to allow a satisfactory periodic motion, ad determine the period. Increase 
the value of « until the motion is aperiodic. Now displace the pendulum an amount a, 
and release it exactly at the beat of a seconds pendulum; determine the deflection from 
its position of equilibrium at, say, every 5 or 10 beats of the pendulum. On substituting 
the values of a,, n, and ¢ in equation (60) we can determine « by trial, each observation 
giving a value of «; the average can then be taken. It would be very difficult to deter- 
