Pa'per on the Length of the Pendulum. 
and, therefore, by introducing this result into the value of in 
equation (3), it will become, 
32 A log (1^ T . ■ 
and the expression for the correction due to the amplitude of 
the arc of vibration, will be 
n{a-Arh) (a — b) 
' 7 ^\ 
32Alog(^^y 
This formula is very similar to that given by Chevalier Bor- 
da mthout demonstration ; for, by only substituting in place of 
the arcs a, b, (a -f b) and {a — b,) the sines of these arcs, we 
shall obtain, 
g — ^ j_ sin (a 4- b) sin {a — ^ which is the formula given for 
1 1 /sina\ ® 
the correction by Borda. 
But when the arcs a and b are given in degrees, the lengths 
of these arcs will be 0,01745329 a, and 0,017453296 b, respec- 
tively, and, therefore, these values being substituted for a and 5, 
in equation No. 4, it will become, 
w (a + &) (a — b) 
105049,57 A log 
n (a + h) (a — b) 
241886,08 
(I) 
By applying our formula to all the finite arcs of vibration, 
we shall reduce them to the case of infinitely small oscillations, 
and by this means we shall be enabled to find the number of 
Injinitely small vibrations which the invariable pendulum would 
have made in the same time. 
Let us take for example the 5th experiment, that is, the ex- 
periment marked E, the great weight being below, since it ap- 
pears to be as free from irregularities in the periods of coinci- 
dences, and in the decrease of the arcs of vibration in geometri- 
cal progression, as any one of the whole set ; but even this is 
' not exempt from them. 
VOL. T. XO. 2. OCTOBER 1819* Z 
