THEORY OF THE SEISMOGRAPH. 189 
in separating displacements and rotations; for their relative effects in producing move- 
ments of the seismograph depend on the arbitrary choice of the axis for the rotation. 
This appears in equation (16), where the values of the various coefficients depend on the 
choice of the origin of codrdinates, about which the rotations are supposed to take place. 
If we could get rid of the effects of linear displacements, we could determine the rota- 
tions. This was first done by Professor Milne,’ who supported a beam by knife-edges 
at its center of gravity; and later by Dr. Schliiter.2 These instruments failed to show 
any tilts at the times of the earthquakes, which therefore must be extremely small. 
A second method of determining tilts has been proposed by Professor Wiechert.® 
Let two horizontal pendulums with equal values of x, nJ/L and gi/L be installed, one 
vertically over the other; and let the origin of codrdinates be chosen at the CG of 
the lower pendulum; its equation will be 
Uh » 5, da, nid, gi/nlo, ' 
Pa dm nde, gi =0 
Tae a Ae ae a) hs 
the equation of the upper pendulum will be 
@a, da, nld@é nlZ' dw, , gi/nlw 
Nea el PP aL) ry aati bck a Le iy ine y : '— 0 / 
ies dea. dee a At i +42) ea = 

it contains an extra term — (nlZ'/L) d’w,/d? where Z' = Z-— il is, in this case, the dis- 
tance between the centers of gravity of the 2 pendulums; the origin of this term will 
appear on referring to equation (16). On taking the difference of these two equations 
we get 
d? (ay — ) 
dt 
d(dyg—,) _ nlZ! Pu, 
9 
oe dt Le oe 

+5 (ay — a) F (po! — py’) = 0 (143) 
which gives us a relation between the record and the angular acceleration of the earth 
about the axis of y without containing the linear displacement. If we work out the value 
of w, and substitute it in equation (25) we can then find the linear acceleration. Prince 
Galitzin has shown a very elegant manner of carrying out this process by the use of his 
method of electromagnetic recording thru a galvanometer.* Professor Wiechert’s method 
presupposes that the supports of the 2 pendulums move as tho they were parts of a rigid 
body, and therefore that the motions can be represented as the same rotation about the 
same axis. This would certainly not be the case if the upper instrument were mounted 
in a high building, for then the vibrations of the building would interfere; and it may 
be questioned whether the condition would hold for two points at different distances 
below the surface of the earth. But if two pendulums are mounted, one above the other 
on the same support, as Prince Galitzin arranged them in his experiments, these objections 
disappear. 
A similar method can be applied to vertical motion instruments; let us suppose that 
two similar instruments are mounted close together with their axes of rotation in the same 
straight line, but with their beams pointing in opposite directions; it is evident that any 
vertical displacement would affect them alike, but a rotation around their common axis 
of rotation would cause movements in opposite directions. The equations of the two 
1 British Assoc. Reports, 1892. 
? Schwingungsart und Weg der Erdbebenwellen. Gerland’s Beitriige zur Geophysik, 1903, vol. V, 
pp. 314-359, 401-465. 
§ Principien fiir die Beurtheilung der Wirksamkeit von Seismographen. Verhand. 1** Intern. Seismol. 
Konferenz. Gerland’s Beitrige zur Geophysik, Ergiinzungsband I, pp. 264-280. 
4 Ueber die Methode zur Beobachtung von Neigungswellen. Acad Imp. des Sciences St. Peters- 
burg. C.R. Com. Perm. Sismique, 1905, T. II, Liv. II, pp. 1-144. 
