75 
of experiments on the pendulum. 
combinations being also 2 n , it is obvious that the fraction 
alone must express the probability of a result totally free 
from error. The neighbouring terms on each side, for 
n = 100, are .078025, .073524, and .066588, the sum of the_ 
7 being .515860 ; and since this sum exceeds it is obviously 
more probable that the result of 100 trials will be found in 
some of these seven terms, than in any of the remaining 94, 
and that the mean error will not exceed When n is so 
large, that the terms concerned may be considered as nearly 
equal, the factors — ' ■, ? . . ., may be expressed by 1 — 
- , 1 — - , 1 — — ..., and the terms themselves by 1, 1 — 
1 — , 1 — . . . the negative parts forming the series 
” (*» 4> 9 • • •) which the sum, for q terms, is-^(y9 3 + T 
q* ^ q) or ultimately j- n q 3 1 consequently if we call the 
middle term e, we must determine q in such a manner as to 
have e (sq - ± q 3 ) = | — e, and q ( 1— q‘) = ~ — i; 
but e has been already found, in this case, = v/^> and ne- 
glecting at first the square of q, we have q — \ V {pi i) 
and ?’ = 7 6 pn, whence j n q' = ^p, and i - i q‘ = .934,55 • 
hence, for a second approximation, *93455 q = - — J-, and 
q = .2674 V {pn) — .53 ; and by continuing the operation we 
obtain .9235 q = ± — 1 , and q = .271 V{pn) — .54; con- 
sequently the probable error, being expressed by -~L j w jll be 
•54 2 V “ — * This formula, for n = 100, 
becomes .0571, and for n = 10000, .00679 — .00011 = 
,00668. 
