THEORY OF THE SEISMOGRAPH. 187 
The form of instrument in most general use for recording vertical motion is that 
developed by Professor Vicentini of Padua,’ though the principle seems to have been used 
in Comrie, Scotland, in 1841.? It consists of a heavy mass supported by an elastic rod 
so that it vibrates in a vertical plane, and records by means of multiplying levers on 
smoked paper. Usually there is no damping. The complete theory of this instrument 
is that of a weighted elastic rod, and is very complicated. It has several proper periods 
of vibration, and it would be set in vertical motion by a horizontal displacement in the 
direction in which it points. We can, however, develop an approximate theory which 








Fig. 62. 
is quite simple. Let the instrument point in the positive direction of y and let z be posi- 
tive upwards; see figure 62. Take the origin O, at a distance ¢ below the point where 
the rod is supported. Later, ¢ will be considered the vertical displacement of the support. 
Let l be the length of the rod, J the so-called moment of inertia of its cross-section, which 
we consider constant; H Young’s modulus for the material of the rod, M the mass at its 
end, and M’ an arbitrary mass. If we consider the bar but slightly bent, its curvature 
will be represented by d?z/dy?; and we have from the general theory of a loaded cantilever, 
neglecting the weight of the rod, 
EJSd2/ dy? = — M'g(l—y) (135) 
Integrate this equation twice, and determine the constants of integration by the condi- 
tions that dz/dy=4, and z=, when y=0; we get the equation 
6 EJ (2 —€— ay) = — M'9 (3 ly’ —7’) (136) 
where z refers to the point on the bar whose abscissa is y. For the CG of M', y=1l, and 
letting z now represent the ordinate of this point, we get 
6 EJ (z—£—al) =2 Mg? (137) 
which gives us the ordinate of the CG when it is at rest, the mass being supported by the 
slightly bent rod. 
To determine the acceleration when the weight is not at rest, we may replace M’ by 
M+M,; and by d’Alembert’s principle, equation (137) will still hold when we replace 
M.,g by the force [M ]d*z/dt”, where d’z/dt? is the acceleration of the CG and where [M]= 
M+ (I'+n,7/''+- + -)/1,? is the effective mass of M and the multiplying levers; this 
is analogous to the value of [J] determined on page 155 and can be found in the same way 
by the consideration of the reactions of the multiplying levers. On making these substi- 
tutions we get for the equation of the moving CG in absolute codrdinates 
6 EJ(2—¢—ol) =—2 Mg? —2 [Mes (137a) 

1 Microsismographo per la componente verticale, G. Vicentini e G. Pacher. Boll. Soc. Sismologica 
Italiana, 1899-1900, vol. V, pp. 33-58. 
? British Assoc. Report, 1842, p. 64. 
