Chap. 7] GRAVITATIONAL METHODS 121 



moves periodically, the distances traveled by both rays change and the 

 fringes shift periodically. If light changes to darkness and back to light, 

 the movement has been one fringe, the phase shift one wavelength, and the 

 displacement of the mirror one-half wavelength, or 0.29 microns (since the 

 wavelength of sodium light is 0.58 microns). The shift of the fringes dx, 

 is expressed in terms of fringe width, x, and all observations are reduced 

 to 5 mm arc (not semi-arc) of pendulum movement. The correction on 

 the pendulum period is determined by measuring periods T and correspond- 

 ing fringe shifts (F) under varying conditions of stability of the pendulum 

 support so that the flexure "coefficient" C/ = AT/AF. Then the flexure 

 correction 



ATnexure = (F) • C/ • « = - • ^^^ • C/ . (7-27) 



X (zee; 



A set of flexure observations with calculations is reproduced in Swick's 

 pamphlet.^" The actual displacement of the pendulum support is very 

 small; for firm ground and pillar it varies from 0.06 to 0.10 fringe width or 

 0.017 to 0.029 microns. 



Twin-pendulum procedures give greater accuracy than the interferom- 

 eter method in the determination of flexure. Observations may be 

 started with the companion pendulum at rest or with both pendulums in 

 opposite phases and identical amplitudes. The second method is more 

 accurate, since theoretically the influence of flexure is completely elimi- 

 nated. Only a small correction remains, because of the impossibility of 

 keeping the amplitude and phase relations of the two pendulums constant 

 throughout the entire observation period. If, in the first method, one 

 pendulum is at rest at the time k ioi2 = and <P2 — (pi = 7r/2), and if at the 

 time t the amplitude 02 of the driven pendulum and the amplitude ai of the 

 driving pendulum is observed, the effect of flexure on period and coin- 

 cidence interval respectively is 



ATnex. = ^ 



ai ir(t — to) 



Annex. - 



2 

 0:2 n 



(7-28a) 



Oil 7r(< — to) ' 



provided the phase difference at the end of the observation period is still 

 nearly 90°. If two pendulums swing against each other on the same 

 support, the flexure corrections for each pendulum on period and coin- 

 cidence interval are, at any instant t,^^ 



^^ Loc. cil. 



" H. Schmehl, Zeit. Geophys., 3(4), 157-160 (1927). 



