342 



EXPLORATION GEOPHYSICS 



are mounted at equidistant intervals (0°, 120°, 240°). Two of the masses m2 and mz 

 are supported on a light aluminum ring and a third mass wii is supported by a light 



aluminum rod above the level of the others. 

 The mass «i2 is very approximately equal 

 to nts, and the combined mass of wi and the 

 rod to which it is attached is equal to wa or 



J«3. t 



The derivation of the fundamental equa- 

 tion of the gradiometer is similar to the de- 

 rivation of the fundamental equation of the 

 torsion balance. The origin of coordinates 

 is taken at the center of the beam sys- 

 tem. The X, y, and z axes are directed 

 toward the north, the east, and vertically 

 downwards respectively. (Figure 195.) The 

 direction of the arm of the gradiometer which 

 carries the upper mass nii makes an angle o 

 with the X direction. As in the case of the 

 torsion balance, the gravity components at 

 a point near the origin are 



-Diagrammatic sketch of the beam 

 system of the gradiometer. 



gz = UxxX + Utvy + UzzS 



gv = UxyX + Uvvy + UytS 



gi = g+ Uxzx + Uyzy + U^zS 



(102) 



The gx and gy components of gravity acting on each mass produce a torque T about 

 the z axis given by 



T= {xgy — ygx) m (103) 



Substituting the values of gy and gx from the system of equations (102) yields 



T = [x-Uxy + xyUyy + xsUyz — yxUxx — y'Uxy — ysUmt] m 

 or 



T= [ ix^ — y^)Uxy + xyU^ + xzUyz — yzUxz\ m (104) 



For the upper mass, .r = / cos a, y = I sin o, and z = — h 



For the lower masses, .r = / cos (a + 120° ),y = l sm (a + 120° ) , and 2 = 



and ;r = /cos (a + 240°), 3; = /sin (a + 240°), and 2 = 



respectively. 



The value of the turning moment due to the circular beam and spokes is zero. 

 Hence, the total torque acting at the three masses is : 



T '= m {P [cos^a — sin^a] Uxy + l^ sin a cos aUn. — Ih cos a Uyz + Ih sin a {/«} 



+ m{/''[cos*(a+120°)-sinMa + 120°)] Uxy + I' sin (a + 120°) cos (a + 120°) C/a} 



+ m{/Mcos*(o + 240°) -sin" (a + 240°)] [/,„ + /" sin (a + 240°) cos (a + 240°) U^} 



or 



T = mP Uxy [cos 2a + cos (2a + 240° ) + cos (2o + 480° ) ] 

 + y2 ml- Ua [sin 2a + sin (2a + 240° ) + sin (2a + 480° ) ] 

 — mhl[Uyz cos a — Uxz sin a] (105) 



But the coefficients of Uxy and Ua are zero as is readily verified by expansion. 



t Compare also A. Broughton Edge and Laby, Geophysical Prospecting (Cambr. Univ. Press, 

 1931), p. 139, pp. 299-300. 



