GRAVITY DETERMINATIONS ON THE CARNEGIE 



73 



observations at sea," by Dr. F. A. Vening Meinesz (5), 

 and also in "The gravity-measuring cruise of the U. S. 

 submarine S-21" (1). The theory and description of the 

 apparatus will be considered, therefore, only in their 

 essentials. 



If all other disturbances except the horizontal ac- 

 celerations are left out of consideration, the equations 

 of motion of two pendulums may be written 



d2ei/dt2 + (g/igi)0i + (1/S.x) dVdt2 = ... (1) 



d2e2/dt2 + (g/£2)^2 + (1/j2 2) d2x/dt2 = ... (2) 



in which d\ and 62 are the angles of elongation of the two 

 two pendulums of length &\ and H2 respectively, g is the 

 acceleration of gravity, and d2x/dt2 is the component in 

 the plane of oscillation of the horizontal accelerations of 

 the knife edges. If the two pendulums are isochronous, 

 then j^i = ^2 = ^^ say, and the result of subtracting (2) 

 from (1) gives 



(d2ei/dt2 - d2e2/dt2) + g/fi (91- 62) = . .(3) 



(3) is just the equation of motion of an undisturbed ficti- 

 tious pendulum of length & and angle of elongation (d^ - 

 62), which is isochronous with each of the original pen- 

 dulums. In the apparatus (d\ - 62) is recorded photo- 

 graphically by means of mirrors attached to the tops of 

 the pendulums. If it were possible to produce two pendu- 

 lums which were isochronous within sufficiently small 

 limits, the problem would be much simpler. Since this 

 is not possible, the effect of any deviation in isochronism 

 between the two original pendulums must be considered. 



If this deviation from isochronism is sufficiently 

 small, the only effect is to alter the period T of the fic- 

 titious pendulum from Ti, the period of the first original 

 pendulum, by an amount 5T where 



and 



T =Ti + (5T (4) 



(5T = -(T2 - Ti) (aa/a) cos (((.2 - *) (5) 



in which a2 and a are, respectively, the amplitudes of the 

 second original pendulum and of the fictitious pendulum, 

 and 2 and <(> their angles of elongation. {T2 - Ti) is the 

 difference in the periods of the original pendulums at the 

 temperature and air density during the observation. The 

 apparatus records photographically (^2 and 0, and since 

 a2 cos (0 2 - <(') is the component of the pendulum-vector 

 of the second pendulum in the direction of the fictitious 

 pendulum-vector, the recording apparatus is arranged to 

 facilitate, during most of the record, the measurement 

 of a2 cos (0 2 - <^)- This accounts for the change in ap- 

 pearance of the middle part of the records in figure 4. 



Effect of Vertical Accelerations 



The effect of vertical accelerations, during the ob- 

 servations throughout an interval t, on the re suiting value 

 of gravity will be simply the average value of those ac- 

 celerations over the interval, that is. 



1/t fo^ (d2y/dt2) dt 



(6) 



which is equal to 



This is to say that the effect of the vertical accelerations 

 on the measured gravity is simply the difference in the 

 vertical velocities at the beginning and end of the obser- 

 vation, divided by the time-interval t. This effect may 

 be made as small as desired in two ways: first by mak- 

 ing t large, and second by taking at the beginning and 

 end of the observation the mean of the positions of the 

 fictitious pendulum-vector for a great many seconds 

 over which time the average value of the vertical veloci- 

 ty may be quite small. In this way observations of half 

 an hour are more than sufficient to make this disturbance 

 negligible. 



Effect of Angular Movements 



If /3 is the angle between the plane in which the pen- 

 dulums swing and the vertical, the component of gravity 

 in this plane is g cos /3. The angle Q is made up partly 

 of a constant quantity /3c and partly of a fluctuating quan- 

 tity a, the period of which depends on the period of the 

 apparatus in the gimbal suspension. The effects of ac- 

 celerations introduced by rotations of the pendulum sys- 

 tem about a horizontal and. about a vertical axis in the 

 plane of the pendulums must also be considered. The 

 correction, 6T, to the period of the fictitious pendulum 

 for the latter effects may be combined with that due to 

 the tilt of the swinging plane. It is given approximately 

 by 



in which 



6T = 1/4 Ti [0c2 + Ca2] 



C = 1/2 [1 - 2(Ti2/Tf2)] 



(8) 



(9) 



1/t [(dy/dt)t - (dy/dt)o] (7) 



and Ti = period of the first original pendulum, Tf = peri- 

 od of the fluctuating tilt, Qc^ and a 2 are mean values dur- 

 ing the observation. The approximation is accurate 

 enough if no one of the periods of the fluctuating quanti- 

 ties is near that of the pendulums. This will be the case 

 if the amplitude of the fictitious pendulums does not show 

 any trace of fluctuations. 



Other Corrections 



The correction to the period because of the finite 

 amplitudes of the pendulums is given by 



6T = 1/16 Ti [{a + 1.5 a2 cos (02 - 0)} 2 



+ a22 - 1/4 a22 cos 2 (02 - 0)] (10) 



in which, as before, Tj is the period of the first original 

 pendulum, a is the amplitude of the fictitious pendulum, 

 a2 is the amplitude of the second original pendulum, 2 

 is the elongation angle of the second original pendulum, 

 and is the elongation angle of the fictitious pendulum. 

 As in the case of the ordinary pendulum, corrections 

 must be applied for the temperature of the pendulums, 

 for the density of the air surrounding them, and for rate 

 of chronometer. These are as follows: 



Temperature: 



Density: 



Rate of chronometer: 



6T=cit 



6T=diD 



6T = -115.7 rTx 10"'' sec 



. (11) 

 . (12) 

 .(13) 



