Chap. 7] GRAVITATIONAL METHODS 179 



These equations indicate that in any azimuth the effect of curvatures 

 is in the same direction and that of the gradients is in opposite direction 

 for both beams. The following set of equations is then obtained for three 

 positions : 



Position 1 



a' = 180°: n[ - n'o = 2&'U^ - h'Uy^ 



a" = 0°: n'l - n'o = 2a"U^ - h"Uy, 



Position 2 

 a = 300°: na - no = -a' sin 60° Ui, - 2a' cos 60° Ury 



+ b' sin 60° U:,^ + b' cos 60° Uy, 



a" = 120°: 712 - no = -a" sin 60° C/a - 2a" cos 60° U^ 



- h" sin 60° U,^ - h" cos 60° Uy^ 

 Position 3 



a' = 60°: n's -n'o = a' sin 60° ^a - 2a' cos 60° U^^y 



- b' sin 60° U^^ + b' cos 60° C/„. 

 a" = 240°: n',' - n'o = a" sin 60° U^ - 2a" cos 60° U^^y 



4- b" sin 60° C/„ - b" cos 60° t/„. 



The two torsionless position readings, no and no , are obtained from the 

 arithmetic mean of the deflections 



I ni + n2 + ng „^ , ji ni + n2 + ng ._ _„,s 



no = ^ and no = ^ . (7-526) 



o o 



Two positions are therefore sufficient to calculate the other four unknowns. 

 Combination of positions 2 and 3 is in most general use, but combinations 

 1 and 3, and 1 and 2 are equally suitable. With the notations ni — no = Ai ; 



/ / . / / / 1 I -y " " ^ II II 11 .11 



n2 — no = A2 ; ng — no = Ag and ni — no = Ai ; n2 — no = A2 ; 

 ng — no = Ag , the equations ior positions 2 and 3 are: 



^ a'f/A - a'C/.„ + ^ ^" 



(1) A2 = —^ a'f/A - a'C/.„ + X^ b'[/„ + WUy. 



(2) A'2 = -^ a"C/A - a"C/., - ^ b"C/„ - W'Uy^ 



^ a'C/A -a'C/., -^^^'^ 



(3) Ag = y^ a'C/A -a'C/,, - X^ b't/^. -\- WUy^ 



^- a"?7A -a"t/^ + ^^»^"^ 



(4) a;' = \^ a"?7A - a"t/^ + ^ b"t/x. - W'Vy^ 



