Chap. 7] GRAVITATIONAL METHODS 197 



inertia K. The period is determined by reading the time required for ten 

 complete swings and repeating five to six times. This gives an accuracy 

 in T of better than 1/100 of a second and a torsional coefficient t = 

 A-ZK/T^. The moment of inertia, K, is determined by calibration with 

 a ring of calculated moment of inertia. If the mass of the ring is m, 

 its outer radius R, and its inner radius r, its moment of inertia is Kr = 

 ^m{R^ — /). If Ti is the period without the ring and T2 with the ring, 



and 



Tl = '^-^{K + Ka) 



T 



from which 



K==Kh . ^' . . (7-60i) 



T\ - Tl 



The torsion coefficient changes with temperature, 9: 



Te = r2o[l + 3(6: - 20°)], (7-60e) 



in which the temperature coefficient, 5> of the .torsional coefficient follows 

 from period observations at different temperatures. In the field the 

 variation of t with temperature is disregarded, but in selecting torsion 

 wires in the factory, wires with excessive temperature coefficients of t 

 are discarded. Equally undesirable are wires of excessive variation of the 

 rest position, n, with temperatui:e. If the reading is rii at a tempera- 

 ture 9i, and n2 at a temperature 62, the variation of n with 9 is defined by 

 112 = ni — a (92 — 9i). The temperature coefficient a of a good wire 

 should not exceed 3-5 • 10""^ nam per degree C (referred to the same optical 

 magnification as used in the field instrument). The temperature coeffi- 

 cient of a wire depends largely on the method of clamping and heat 

 treatment.*^ Platinum wires are heat treated electrically under load; 

 tungsten wires are annealed in an atmosphere devoid of oxygen. Fig. 7-69 

 illustrates the improvement brought about in a wire by heat treatment. 

 Curve A is the zero shift of an untreated wire, curve B that of a wire 

 tempered under load fifty times a day. 



It should be noted that in a test instrument the change of the zero 

 position with temperature is generally not the same as in a torsion balance 



" Concerning the effect of annealing and baking on temperature coefficient, see 

 N. N. Zirbel, Physics, 2(3), 134-138 (March, 1932). 



