for determining the Length of the simple Petidtihwi. 99 



the same side as it, with respect to the centre of gravity, be 

 denoted by 5. In oi'der to avoid confusion, I shall always sup- 

 pose that the pendulum, when referred to, is placed with the 

 same end uppermost; by which means we may, in all cases, 

 denote the three axes, above alluded to, by the desi(>nation of 

 7ipper, middle, and lower axes: and each of the four combina- 

 tions of three knife edges will give the length of the simple pen- 

 dulum, agreeably to the formula of M.Prony, which is as follows. 

 By determining the number of vibrations, made in a given 

 time, by each of the knife edges, we may obtain for the upper, 

 middle, and lower knife edges respectively, the quantities ft', 

 ii"f n'" ; each of which represents a quantity {n) of the form 



— rr- ) : where T denotes the mean solar time (expressed in 



seconds) employed in making N vibrations; and tt the cir- 

 cumference of the circle, diameter equal to unity. If T re- 

 present a mean solar day ( = 864'00 seconds), N will conse- 

 quently denote the number of vibrations made in a mean solar 

 day : the value of N being always supposed to be corrected 

 for the magnitude of the arc, the expansion of the pendulum, 

 the reduction to a vacuum, and the rate of the clock. Either 

 of the above quantities (?^', n", n'") thus deduced, being mul- 

 tiplied by the accelerative force of gravity [g), will give the 

 length of the pendulum synchronous with a pendulum corre- 

 sponding to its respective axis: and by multiplying this length 



by ( ortQn ) ^® obtain the length of the simple pendulum vi- 

 brating seconds of mean solar time, at the given place. 



The value of ^^ is determined by means of a quadratic equa- 

 tion in the following manner. Make 



v' = «'" + n' 

 v" = n'" + n<< 

 ^11 = n» - n' 



« = 2(8v' + Av'") 



i = S-v' + 2 8Av" + A^v'" 



c = tA {An" + 8?i"') 



Then will the distance (x) of the upper axis from the centre 

 of gravity, the accelerative force of gravity (g), and the mo- 

 mentum of inertia (ju,), be found from the following equations, 



^ = IT ± vc^y - -V 



_ S(2i— J) _ A(2x -A) 

 ° 7i" i — t"'x S J- — A n'" 



fx. = X (g n'—x) 



2 And 



