128 THE ROYAL SOCIETY OF CANADA 



the distance between the knife edges), it follows that the moment of 

 inertia of the pair of weights about either knife edge is 2m{k-+x- 

 +XV4). 



/, /', are the moments of inertia of the remaining parts of the 

 pendulum about the respective knife edges. M is the mass of the 

 whole pendulum and h, L — h the distances of the knife edges from 

 the centre of gravity of the whole. /, /', and // are thus independent 

 of the position of the sliding weights and therefore of x. 



The periods for any setting of the adjustable weights are therefore: 



r = 27rV{7+2w(P+x-+LV4) \/Mgh 



and r = 2TrV \r-}-2m{k'-+x'-\'L^/ 4:) }/Mg{L-h) 



and hence the cur\-es T" against .x-, T''^ against x~ are rectilinear. 

 The equality period may therefore be found from four observations 

 of period, two with the sliding weights at a position Xi and two at a 

 position .To, either by setting out the points on coordinate paper, 

 joining the corresponding pairs of points by straight lines and finding 

 the period corresponding to the point of intersection, or by d'rect 

 calculation. If Ti, 7\', are the periods for the position xi, and T2, 

 T2', for the position X2, it may be readily shown that the formula 

 for calculating the required period is 



7V= { T{^T'.,--T^T\^\ ! { (rV- r'l-) - {T.^- 7V-) j • 

 5. Reduction Factors. 



In order to obtain the value of any T" from the observed value 

 the square of the period, P, the value of P' must be multiplied by the 

 following factors: — 



(1—— ) , on account of the finite amplitude, a, of the 

 V 16/ 



angle of swing, 



( 1 — ), to correct for damping for which X is the logar- 

 ithmic decrement, 



( 1 — — ) , where ill is the moment of inertia of the hydro- 

 dynamic mass of the air set in motion by the pendulum compared 

 with the moment of inertia of the pendulum itself, 



on account of the buoyancy of the mass m of 



mL, \ 



2MÏi/' 

 air displaced. 



(' 



