OF FLUIDS ON THE MOTION OF PENDULUMS. [89] 



arc at the beginning to the arc at the end of an oscillation, we must put t = t in (168), 

 whence, neglecting the effect of the wire, we obtain 



■ wft cr , - . 



l -Y-s (169) 



If now A ft' be the correction to be applied to k' in this formula on account of the wire, 

 since k', ft/ are combined together in the expression for the arc just as ft, k i in the expression 



for the time, we get 



ft' 



Aft' = -J- Aft, (170) 



fti 



and the approximate formulae (115) give 



4L 

 Aft = Aft, (171) 



7T 



whence the numerical value of Aft' is easily deduced from that of Aft, which has been already 



calculated. We get also from (52) 



ft'=ft-l+|(ft-l) 8 (172) 



whence ft' may be readily deduced from ft, which has been already calculated. 



74. Before comparing these formula? with Bessel's experiments, it will be proper to 

 enquire how far the latter are satisfied by supposing the arcs of oscillation to decrease in 

 geometric progression. In Bessel's tables the arc is registered in the column headed /m. 

 This letter denotes the number of French lines read off on a scale placed behind the wire, 

 and a little above the sphere, and is reckoned from the position of instantaneous rest of the 

 wire on one side of the vertical to the corresponding position on the other side. The distance 

 of the scale from the axis of suspension being given, as well as the correction to be applied 

 to /n on account of parallax, the arc of oscillation may be readily deduced. However, for 

 our present purpose, any quantity to which the arc is proportional will do as well as the 

 arc itself, and fx, though strictly proportional to the tangent of the arc, may be regarded as 

 proportional to the arc itself, inasmuch as the initial arc usually amounted to only about 50' 

 on each side of the vertical. 



Now we may form a very good judgment as to the degree of accuracy of the geometric 

 formula by comparing the arc observed in the middle of an experiment with the geometric 

 mean of the initial and final arcs. I have treated in this way Bessel's experiments, Nos. 1, 2, 

 3, 4, and 5. Each of these is in fact a group of six experiments, four with the long pendulum 

 and two with the short, so that the whole consists of 20 experiments with the long pendulum, 

 and 10 with the short. In the case of the long pendulum, the observed value of fx regularly 

 fell short of the calculated value, and that by a tolerably constant quantity. The mean differ- 

 ence amounted to 0.688 line, and the mean error in this quantity to 0.109. This mean error 

 was not due entirely to errors of observation, or variations in the state of the air, &c, but 

 partly also to slight variations in the initial arc, larger differences usually accompanying larger 

 initial arcs. The initial arc usually corresponded to y. = 39 or 40 lines, and the final to /u = 15 

 Vol. IX. Part II. 36 



