of experiments on the pendulum. 
73 
n — 2 
» = 3 
» = 4 
?z — 6 
Coefficients 
1 2 1 
* 3 3 
1 1464 
I 
i 6 15 20 . . . 
Numbers thrown 
024 
024 
6 0246 
8 
0 2 4 6 . . . 
Differences from n 
202 
3 1 1 
3 4202 
4 
6 4 2 0 . . . 
Errors of the means 
1 0 1 
1 £ x 
1 S 3 
1 1 i 0 f 
1 
x | i 0... 
Sums of errors 
I+0+I=2 
i+i+i+i=4 1+2+0+2+1 
=6 1+4+5 + 0 . .. 
Mean errors 
i — 1 
4 — S 
4 1 
T— T 
6—J. 
T6 — 3 
2 O ..S 
T T — TS - 
n — 8 
1 8 28 56 70 
02468 
86420 
1 I i i o 
1 + 6+144.14+0 
7 0 _ 3 5 
TT 6 — 12T 
It is easy to perceive that these coefficients must express 
the true numbers of the combinations, since they are formed 
by adding together the two adjacent members of the pre- 
ceding series; thus when n is 3, 1 combination giving the 
number o and 3 the number 2, these two combinations, being 
again respectively combined with 2 and o of a fourth counter, 
give 1 3 = 4> for the combinations affording the number 2 
in the next series ; while each succeeding series must continue 
to begin and end with unity, since there is only one combina- 
tion that can afford either of the extremes. 
In order to continue the calculation with greater conve- 
nience, we must find a general expression for the middle 
terms, 2, 6, 20, 70 . . neglecting the odd values of n. The 
first, 2, is made up of ( 1 -{- 1 ), the second, 6 , is 2 ( 2 + 1); 
20 is 2 (6+ 4) and 70 = 2 (20 15): or 6 = 2 (2. J-), 20 = 
2 (6. f), 70= 2 (20. J), whence the series may easily be 
continued at pleasure, multiplying always the preceding 
term by f, i-2, i* UL ... We have also 6=16. i == 2* . 
20=2*. and 70 = 2 8 . - 3 2 5 q- : consequently the terms of 
this series, divided by 2*, will always express the mean errors 
already calculated. From this value of the middle term we 
may easily deduce that of the neighbouring terms by means 
of the original formula n . — 
MDCCCXIX. 
+ M + I 
+ n 
in n + 1 in+ 2 
z • • ; the 
