178 GRAVITATIONAL METHODS [Chap. 7 



less position) the reading corresponding to zero deflection, w — no = 2<pt 

 (or 4:<pf with double reflection, as in the small "Z"-bar and tilt-beam 



Askania balances). Then, n — no = 2f , or 



fK . - /d'U d'U\ ,iK ^ _ d'U 

 no = — sin 2a I —: - z-^) + - cos 2«.2 — — 



, 2imlh d'U 2fmlh d^U . 



H cos a — sm a 



T dydz T dxdz 



)■ (7-51a) 



This is the principal equation of the torsion balance of the second kind. 

 In the' abbreviated notation previously referred to, and with the "instru- 

 ment constants,"" 



iK ^ , 2fmlh 

 a. = — and b s , 



r T 



the principal equation (see Fig. 7-60) is 



n — no = a(C/A sin 2a + 2Uxy cos 2a) -\- b(cos aUyz — sin alJxz). (7-516) 



This equation contains five unknown quantities: the four second deriva- 

 tives of the gravity potential and no. They may be determined from five 

 equations by observing beam deflections in five azimuths (0°, 72°, 

 144°, 216°, and 288°). Such observations take considerable time, owing 

 to the long period of the beam. Since T = 2ir \/K/t, a beam with 

 K ^ 20,000 C.G.S. units and t ^ 0.5 C.G.S. unit has a period of the 

 order of 20 minutes. To reduce the observation time, Eotvos designed a 

 double instrument with two beams in antiparallel arrangement, rotated 

 together into the respective azimuths. This adds a sixth unknown 

 (the torsionless position of the second beam). However, since two 

 observations are made in each azimuth, six quantities may be ob- 

 tained in three positions. 



3. The double gradient and curvature variometer is usually so arranged 

 that when the photographic or reading device is oriented toward north, 

 the hanging weight of balance I is in the south, and a' = 180°, a" = 0. 

 Then a'b' are the constants for the first balance and a"b" for the second 

 balance. The subscripts 1, 2, and 3 refer to the azimuths (0°, 120°, 240°). 

 Then the general equations are, for beam II (az. a), 



n" — n'o = a" (sin 2a- (7a + cos 2a-2Uxy) -\- h" (cos a- Uyz — sin a- Uxz) 



and, for beam I (az. a -f- tt), 



n' — n^ = a' (sin 2a -Ui, -\- cos 2a-2f7i„) — b' (cos a-Uyz — sin a- Uxz)-, 



(7-52a) 



'''' The above symbols are customarily used for the instrument constants and are 

 not identical with the quantities a and b defined by eq. (7-486). 



