ANOMALOUS BEHAVIOUR OF DELICATE BALANCES, ETC. 397 



concave mirror was secured to either end of the beam, and the lamp and scale arranged 

 as in the former case. 



Two series of experiments were made : the one with the Becker, and the other 

 with the circular beam. Having attached the mirror to the extremity of the L arm 

 of the beam, various loads were placed in the pan, and the resulting deflections noted. 

 The mirror was now removed from the L arm and secured to the extremity of the 

 R arm. Reversing the beam in order to interchange the positions occupied by the 

 terminal knife-edges, the experiments with the several loads were repeated. For 

 equal loads the values obtained for the flexure of each arm were in every case 

 identical. 



Similar experiments conducted with the circular beam also gave concordant results 

 for the two arms. From these experiments we conclude that if the arms exhibit a 

 differential flexure when the beam is loaded, its limiting value is less than 0'0004 mm. 

 If, however, there is any difference in the flexure, the effective length of the arm 

 which bends to the greatest extent will be less than that of the more rigid one ; and 

 consequently the R.P. will be shifted in the direction of the shortened arm. 



It can be shown that the accuracy of weighing is not appreciably affected even 

 when a possible inequality in the rigidity of the arms results in a difference in the 

 depressions equal to - 0004 mm. 



I am indebted to Mr. H. H. HILTON, formerly Fellow of Magdalen College, Oxford, 

 for the following proof of the accuracy of this view. 



Let us assume the arms to be of equal length, 70 mm. Suppose that when the 

 pans are loaded with weights, W, w, the arms are depressed through small angles, 

 a, fi, respectively, from the horizontal, owing to a lack of perfect rigidity in the arms. 



Then, if equilibrium exists, 



W cos a = w cos ft, 

 therefore 



W w = W (1 cos a sec ft) 



= ^-W (a ft) (a + ft) approximately. 



Now, in the experiments described above, the difference of depression of the two 

 ends of the beam was never more than 0'0004 mm. even when W and iu = 200 gr. ; 

 therefore (a ft) was never more than - 0004/70 = 0'0000057 ; while ( + /?) was never 

 more than O'OOG. Therefore Ww was never more than 1 x 200 x 0'0000057 x O'OOG 

 = 0'000004 gr., which is a negligible quantity. 



After reviewing the whole of the available evidence, we were led to conclude that 

 although a difference in the flexure of the two arms of a loaded beam is possible and 

 probable, yet its magnitude, even in the most marked case, is so insignificant that it 

 could produce no appreciable error. 



Dismissing, therefore, the possibility of the introduction of any significant error 

 from this cause, we must seek for some other explanation of the observed changes 



