THEORY OF THE SEISMOGRAPH. 153 
The rotations are so small that we may neglect the order in which they are effected, and 
their coefficients may be considered constants. VSS eas we get 






Chee as w dw, 1 “6 
a — il —L—(¥+1 
dt? =o a+ 2 aa RAT!) de ae 
ay _ dn dw, Pe 
dt? me fies: dt? an) 
(a Puy 
— — X —4 
dt? =] — dt? 

Introducing these values in equation (12), and the value of F’, thus obtained in equation 
(9) and writing J, + M/l’?= J, the moment inertia around the axis of rotation, we get, 
GO, , Po, ae ; a tae ( # 
ii 3— Mi — Z—il rc a 1 — 
to ge tt ae be ari Aah Bersted Gs 05 dt? “er 4 




lee’ (Z Tat +) (16) 
dt? dt? 
sry Wy, 
fap +0 Ger x Gao} (+8) 
The force f is small; and on account of friction between the pendulum and the indicator, 
its direction is not accurately known, but as the friction and the angular displacements are 
small, it is nearly at right angles to both the pendulum and the indicator; we have there- 
fore replaced /, in equation (9) by f, (1—6°/2) and have neglected the term f, (w, + 6) 
in obtaining equation (16). This is the general equation of the horizontal pendulum 
seismograph, within the approximations mentioned. The successive lines of the second 
member give the moments around the axis (3) due to forces parallel to the axes of z, y, 
and z respectively; (the term M/d?0/di*, which has been combined in the first term of 
the first member, should be restored to the first line of the second member to make the 
statement strictly true) and the origin of the force represented by each term in the equa- 
tion is evident. 
This equation can be greatly simplified by a proper choice of the origin of codrdinates ; 
if we place the origin at 0’, we have X = Y = Z =0, and the equation becomes 
#60, , &w (aE Go, fil Y 
T,—*8 = M1| )22_ it iS ee 
to qe t ag | dt? at att aa ( 2 
(ae ta} OO Lie He +9} (B+ 8) 
On putting J,=0, omitting 6°/2, f,), and lw,d’w,/di?, and making the proper changes of 
notation, this becomes the equation No. 86 of Prince Galitzin.t! Equation (16) can be 
simplified still more by placing the origin at CG,; then A=Y+l=Z—iU=0, and it 
becomes 
a6 Pus PE _ fb\(,_%\ , &y en 
19 ag dt? +s dt? seis le a (1 At dt? wet 8) — cae 9)i oa a) ee 
It is evident that this equation can not be simplified further without omitting some of 
its terms. Referring to the equation of a wave, equation (1); differentiating twice with 
respect to ¢ we find for the maximum value of the acceleration 
TY (maz) = 4 (19) 









(17) 

a 



1 Ueber Seismom. Beobachtungen. Acad. Imp.d. Sci. St. Petersburg. C. R. Com. Sismique Perma- 
nente. 1902, Liv. I, p. 142. 
