OF FLUIDS ON THE MOTION OF PENDULUMS. [91] 



Now, as has been already observed, the arcs of vibration decrease nearly in geometric pro- 

 gression. If this law were strictly true, we should have 



t 

 a 



© T '. <"" 



where a denotes the initial and a 2 the final arc, and T denotes the whole time of obser- 

 vation. We may, without committing any material error, substitute this value of a in 

 the last term of (174). The magnitude of the error we thus commit is not to be judged 

 of merely by the smallness of B. The approximate expression (175) is rather to be 

 regarded as a well-chosen formula of interpolation, and in fact T~ x log 6 (a a 2 _I ) differs 

 very sensibly from A. Making now this substitution in (174), integrating, and after inte- 

 gration restoring a in the last term by means of (175), we get 



BTa , _ 



loga=-^-, -. +C, (176) 



log a 2 - log a 



C being an arbitrary constant. To determine the three constants A, B, C, let c^ be the 

 arc observed at the middle of the experiment, apply the last equation to the arcs a , «,, c^, 

 and take the first and second differences of each member of the equation. Let A! denote 

 the sum of the two first differences, so that Ajtf is the same thing as T. Then we may 

 take for the two equations to determine A and B 



A, loga = - A A, t — ; A 2 log et = -^ . 



A, log a„ A' log a 



Eliminating B, and passing from Napierian to common logarithms, which will be denoted 

 by Log., we get 



= -A.Logq, f _ A»Logq .A,a. l 



Loge.A,* 1 A, Log « . A 2 aj' ' V ' 



da. 

 If we suppose the part of — — which does not varv as the first power of a to be 



CLZ 



a 2 (p'(a) instead of Ba", we shall get in the same way 



J = -A,Logq,, f A 2 Loga o .A 1 0(a o ) j 

 Loge.A^ 1 A 1 Loga„.A 2 ^(a )J* 



76. I have not attempted to deduce evidence for or against the truth of equation (173) 

 from Bessel's experiments. The approximate formula (175) so nearly satisfied the obser- 

 vations, that almost any reasonable formula of interpolation which introduced one new 

 disposable constant would represent the experiments within the limits of errors of obser- 

 vation. It may be observed, that the factor outside the brackets in equations (177) and 

 (178) is the first approximate value of A got by using only the initial and final arcs, 

 and supposing the arcs to decrease in geometric progression. In the case of the long 

 pendulum, the value of A, corrected in accordance with the formula (178), would be very 

 sensibly different according as we supposed (p(a) to be equal to Ba, in which case (178) 



36—2 



