331 
Paper on the Length of the Pendulum. 
Let us suppose, therefore, that when the semi-arc of vibra- 
tion is equal to a, the pendulum is left to vibrate in air. Then, 
since the arc a will be continually diminishing, it will successively 
become a', a!\ a'", a' 
a a 
or -, -2 
a a 
/]f 
Cl -L 1 • 
— ; - being 
the common ratio of this decreasing geometrical progression ; 
consequently r must, in this case, represent a number greater 
than unity. Thus each term of the series may be expressed in 
a function of the first term a, and the corresponding number of 
small vibrations, counting from the time when the semi-arc of 
vibration was = ct, will be, 
^ 16 ’ ^ 16 ’ ^ TB ’ ^ 16 ’ 
or. 
1-H 
16? 
:2’ 
or 
1 4. __ 1 4 ^ 1 4 ^ 
he. 
so that after the lapse of n finite oscillations, the corresponding 
number of infinitely small oscillations will be the sum of all the 
preceding terms. Call this number then we shall have. 
7 i' — n-\- 
{±) 
16 
^ , if f ^ {a"J 
16 
or, 71 4 
16 r^ 
4 
16 
+ .... 
“T6~ 
0)' 
16r^ 16?'*^ ‘’T6r2«’ y 
And as the terms of these two values of n\ omitting the first 
term n in each, form a geometrical progression, they may be 
summed ; so that if this sum be represented by S, we shall have, 
S-— • 
“16 
( 2 ). 
(r^ l).?'2w 
We may eliminate r from this value of S, and only leave in 
it the arcs a and i, the first and last terms of the series a, a', a'\ 
— h ; for if we compare the corresponding terms of 
the two identical equations No. 1, we shall find that after n finite 
oscillations have elapsed, there will exist this relation between 
_ o 
them, (a(^)) 
or 
— 
(a («)) 
— f 
wlience, 
