Chap. 7] 



GRAVITATIONAL METHODS 



18{ 



are spaced at angles of 120** (Fig. 7-61). All masses are equal and 

 balanced. Since the moment of the curvature forces on each mass is D2 = 

 mpF (p = radius arm), the sum of the three moments to be used in eq. 

 (7-506) would be zero for any value of a. This may be demonstrated 

 readily for the position a = 0: 



D2 4- D;' + Dr = ? mp'{ C/a sin 60° - Ut, sin 60° 



+ 2U:^ - 2U^y cos 60° - 2U:,y cos 60°} = 0. 



Therefore, formula (7-516) for the standard balance reduces to 



n — no = b (cos a Uyz — sin a Uxz), (7-55) 



where b = 2fmph/T and which contains three unknowns. These may be 

 determined in three azimuths so that when 



a = 0°, 

 a = 120°, 



ni — no = hUyz; 



nz 



-no = ^ ( "2 Uy' ~^^' 



a — 240°, 713 — Wo 



-M 



,v,*^ii. 



•)■ 



and 



Hence, 



f/x. = 



VSb 



\{n% — rio) — (w2 — no)] and 



Vyt = r{nx — no) 



where 



(7-556) 



no = lini + n2 + ns). 

 For four positions (0°, 90°, 180°, and 270°) the readings become 

 ni — no = hUyz, ns — no = — bt/„, , 



n2 — no = —hUxz, n4 — no = hUxz, 



so that ns — no = no — ni and n2 — no = no — n4 . The no is thus half 

 the sum of the opposing readings. Then 



Uxz = su [(^4 - no) — (n2 — no)] 



Uyz = — [(ni - no) ~ (n3 — no)]. 



(7-55c) 



