GRAVITATIONAL METHODS 321 



The light ray, when the balance beam is deflected, reflects from this mirror 

 to another mirror on the side of the tube that houses the torsion wire. This 

 brings in a double reflection so that for the Z-beam balance, we have 



n — iio = 4 a f (73) 



Note : The Uo value or torsionless position is not shown as an actual dot 

 on the photographic record. It is obtained by calculation, and is used as a 

 reference point from which to figure beam deflections. However, it could 

 theoretically be represented by a dot at scale position Hq on the photo plate. 



Fundamental Equation of the Torsion Balance 



Combining Equations 70 and 72 gives : 



T 



Substituting in this equation the final values for Mi and M2 developed 

 in Equations 48 and 69, we have the fundamental equation for a single beam 

 torsion balance^ as : 



fl . ^ rr , fl o OTT . 2finlh J.J 



n— no = - — sm 2(/) c/a + ""^ — cos 2^ 2U j^y H — cos ^Uyz — 



T T T 



2fmlh . „- 



— ?,\n^Uxs (75) 



T 



Definitions of the terms in the equation are Hsted in Table 8. Where these quantities 

 have definite dimensions, those given are for Askania Balance No. 529, a Z-beam instru- 

 ment and a fairly early model of that type. 



TABLE 8. 



SYMBOLS IN FUNDAMENTAL EQUATIONS 



n = the scale reading representing the rest position of the balance beam in 



3/2 mm. scale divisions. 

 Ho = the torsionless position of the balance beam (or undeflected position). A 

 calculated scale reading value. This quantity is an unknozvn in the equa- 

 tion (1). 

 / = the focal length of the lens in the optical system, in 3^ mm. scale divisions. 

 For photographic recording this is 46 cm. (or 920 J^-mm. divisions) in the 

 Z-beam instrument. 

 / = the moment of inertia of the suspended system = 18,586. 

 r ■=. the torsion coefficient of the torsion wire = 0.597. It represents the force 



necessary to twist the wire through a given angle. 

 <p = the angle of orientation of the beam box in degrees from the north. In effect, 

 it is the orientation of the beam. 

 U a. ^= the N-S component of the curvature quantity (2). 

 2Uxv = the E-W component of the curvature quantity (3) 

 These ttvo quantities are iinknoivns. 

 m = the mass on the ends of the beam : 12.0 gr. on the upper end of the Z-beam 

 and 10.6 gr. on the lower end. "m" enters the formula as the sum of these 

 masses or 22.6 gr. 



t Some of the original torsion balances had only the one beam. 



