ESTIMATION OF SMALL EXCESSES OF WEIGHT. 25 



If Q = d, we have the positioa of equilibrium given, by 



ap 



2Fh + Mgk' 

 The semiperiodic time is 



(2) 



Mr+ — 



From equations (2) and (3) we can eliminate 2PA -|- Mgk, 



obtaining 



P = '' ag t^ (4) 



From this expression it appears that, if we know the 

 moment of inertia of the beam, its length, and the weight 

 at each end, we can find the excess p from the time of 

 vibration and deflection. 



The results given in this paper were obtained with a 

 16-inch chemical balance by Oertling. The exact length 

 of the half beam (a) measured by a dividing-engine is 

 20*2484 centimetres. 



To find the Moment of Inertia MI^ of the Beam. — The 

 simplest way theoretically would appear to be this. Find 

 the times of vibration t^, t^, and the deflections fl„ fl^, due 

 to the same excess p with two diff'erent loads Pi, P^ in each 

 pan. Equating the values of p given for the two by 

 equation (4) we have 



an equation which will give lAgV in terms of known 

 quantities; but on trial it was found that a very small 



