GRAVITATIONAL METHODS 



317 



-^U. . 



Z^U 



using cos (90° — m) = sin n and Equations 52 and 53. 



(63) 



From Figure 179 it is seen that the coordinates of m, namely x" and y, are given by 

 x' =■ I cos M and y' = / sin fi. Substituting these values and the value of F from Equa- 

 tion 63 into Equation 61 gives : 



Fig. 179.- — Showing force components acting on a simple tor- 

 sion balance beam as in Figure 178, but with new system of 

 coordinates with x axis along the beam and y axis at right angles 

 to it. The beam makes the angle fi with the x' or minimum curva- 

 ture direction of the equipotential surface. 



Ma = 2ml I — - • I cos fi sin /* — -— — - / sin u cos « ) 

 M. = mP2sm,.cosM(^^^--^j 



(sin 2/4 = 2 sin A cos A) 



M2 = mP sin 2 jit 



?)x'^ 



mi 



2Pm ■=■ I the moment of inertia of the beam ; hence, 



M2— — I sm 2 fi [ ^^-^ — — - I 



2 V ^'^ ^y / 



(64) 



From Figure 179, substituting for the angle m its equal, the angle (^ — \) in Equation 

 64 and noting that 



sin {A -{■ B) — sin A cos 5 — cos .^ sin B 



i- 



\dx'^ 



M2 = I (Vz sm2 <p cos2\ - y2 COS 2 cp sin 2 X) ( |^ - ^ j 



(65) 



