184 REPORT OF THE CALIFORNIA EARTHQUAKE COMMISSION. 
This form was originally suggested by Professor Ewing,’ and the second type above men- 
tioned is called “ Ewing’s duplex pendulum.” A rod attached to the upper pendulum 
records on smoked glass through a multiplying lever, usually multiplying four times. 
The glass does not move and there is no arrangement for recording the time. The record 
of movement is superposed upon itself and is usually difficult to interpret. Several of 
these instruments were working at the time of the California earthquake, and their dia- 
grams are reproduced in Seismograms, Sheet No. 3. 
Lately, Professor Wiechert has greatly improved the inverted pendulum.’ He has 
made it very heavy, 1000 kg. or more, in order that he might magnify the motion sev- 
eral hundred times and still not have the movement too much affected by the solid fric- 
tion of the indicator. He has added a strong viscous friction so as to damp out the proper 
period of the pendulum and has thus produced a very efficient instrument. 
To keep the pendulum in stable equilibrium, springs are attached to a point of the 
pendulum distant /, from its point of support. The forces thus introduced are propor- 
_ tional to the displacement; let us represent these forces brought into play by positive 
angular displacements, 6, and @,, around the axes (1) and (2) respectively, by 21,0, 
and — v,1,0,; v, and v, would in general have about the same values. 
The equations of linear accelerations become 

2 2 
Mo =F +f.— vd, ML= P+ f+ wl, MS =F,- Mg (121) 
The moments become 
The cosines of the angles between the moving and fixed axes are the same as for the 
vertical pendulum, equations (108) and (109). The values of the codrdinates of the CG 
(x, y, 2) are also the same as those given in equation (110), with the sign of J reversed. 
Carrying thru the same operations as before, making the original position of the CG the 
origin of coordinates and omitting the negligible terms, we arrive at the equations 
6, 
—'= Ml —({(tt+—q)\6,' —f 
dt? ie 8 te “) ; Pah 
a6, ae vd, 
yf A it nf es fe 
Jey Tg = — MATE — go, + (9 tah 
Iq) 
(123) 
If there is no disturbance d’n/dé?, d?&/dt’, w, and @, are all zero, and in order that the 
equilibrium should be stable, we must have v,1,?/Ml>g, and v,12/Ml>g. Introduc- 
ing the values of f, and f, from equations (119), dividing by [J,.)], [J], and writing 
UpVM1= L,, [I /MI= L,, we find 
: 2 ,72 2 12 Vo 2 
Se ae P= 0 a 2k es (Hr 7) e=0 (124) 

af I,d@" i, Mi L, df Tae mviguew Mh 7 GL 
After adding damping and frictional terms to these equations, they differ from equation 
(25) only in some of the signs (which is a matter of notation), and in the factor multiply- 
ing the angular displacement. If we replace (v,l,?/Ml—)/g of equations (124) by 7 
they become equivalent to equation (25), and on passing to the marking points, we get 
equations equivalent to (26). Therefore all the characteristics of the horizontal pen- 
dulum and the interpretation of its record may be applied to the inverted pendulum if 
we suppose 7 in the former to be replaced by (v,1,7/M1— q)/g. 

Transactions Seismological Society of Japan, 1882, vol. V, p. 89; and 1883, vol. VI, p 
? Ein astatische Pendel hoher Empfindlichkeit zur mechanischen Registrierung pas eae eben 
E. Wiechert, Gerland’s Beitrige zur Geophysik, 1904, vol. VI, pp. 435-450. 
