Chap. 7] GRAVITATIONAL METHODS 101 



spends to a change in gravity of 1 to 2 milligals. This inverted pendulum 

 is therefore 1000 to 2000 timos more sensitive than the ordinary 

 gravity pendulum. 



2. Pendulum on moving support. The theory of the pendulum on mov- 

 ing support is of equal importance for gravity measurements on vessels 

 and floats and for land observations in connection with the elimination of 

 the flexure of the pendulum support. Details of the theory are given in 

 two publications by Vening Meinesz; only the principal formulas are 

 discussed here. On a moving support three factors alter the period of a 

 pendulum: (1) horizontal accelerations, (2) vertical accelerations of the 

 suspension point, (3) rotational movements of the apparatus. 



When rotational movements are kept down by suspending the apparatus 

 in gimbals, horizontal accelerations cause practically the only interference 

 with the movement of the pendulum. This interference may be com- 

 pletely eliminated by swinging two pendulums simultaneously on the same 

 support in the same vertical plane. By extension of eq. (7-166) the 

 equations for two pendulums may be written 





j2 n j2 , 





(7-18o) 



where y is the horizontal coordinate in the plane of oscillation of the 

 pendulums, m and on their angular frequencies, 6i and 62 their amplitudes, 

 and li and h their lengths. When an optical arrangement is provided 

 whereby only the differences in the amplitudes of the two pendulums are 

 recorded, the following equation is obtained for two isochronous pendulums 

 (wi = 032 and Zi = Z2): 



This relation is identical with the equation of motion of a single undis- 

 turbed pendulum. It holds for a "fictitious" pendulum with the elonga- 

 tion ^1 — 62, the same length I and the same frequency oj as the original 

 pendulums. . A correction is require^ if the two pendulums are not iso- 

 chronous. Denoting the period of the fictitious pendulum by T, that of 

 the first original pendulum by Ti and that of the second by T2, the devia- 



''1 F. A. Vening Meinesz and F. E. Wright, "The gravity measuring cruise of the 

 U. S. Submarine S 21," Publ. U. S. Naval Observatory (Washington), Vol. XIII, 

 App. I (1930) ; F. A. Vening Meinesz, "Theory and Practice of Pendulum Observa- 

 tions at Sea," Publ. Netherlands Geodetic Comm. (Delft, 1929). 



