450 O.N, Rood—Application of the Horizontal Pendulum to the 
Differences. M’. 
B-—A =a+d 
B—A’ =a—d ages 
B/—A’ =2+4+d é 
B’—A”=a—d x 
The average of each pair of differences will give the true 
quantity, a, free from errors introduced by motion. In the 
case now under consideration the average of the column M’ 
will be equal to the average of the column differences, so that 
the final result will be identical whether we employ the = 
of one or the other, but by employing the column M’ we ha 
before us the identical observations free from pa eb He a 
due to the independent motion of the pendulum, thus enabling 
the observer more readily to judge of the reliability of the ob- 
servations, and to calculate their probable error. 
2. The distance passed over independently by the pendulum in- 
creases during each obs-rvation by a constant quantity.—Retaining 
the same notation, etc., as before, we have: 
Readings. 
AeA ae A. 
B =A+a4+d =A+ax+d 
A’ =A+d+4+2d =A  +3d 
B’ =A+a+d+2d+3d =A+a+6éd 
A” =A+d+2d+3d+4d =A  +10d 
BY =A+a+d+2d+3d+4d+id =A+x+15d 
Taking, then, the differences according to the method above 
indicated, we have: 
Diernces M’. M”. M. 
a 
; i d d 
a— 2d L— - are a 
x+3d 
a a 
t 2 2 
e+5d 
The column marked M’ is obtained by taking the average of 
irs of differences as indicated by the vinculums on their left- 
and side; the column M” by using those on the right; the 
final column M. by sorabinihi’ in pairs the quantities under M’ 
and M”. It is evident that the final column of means, M, will 
it this case give the correct value, or a, and that the average of 
the column of differences will be incorrect by a if an even 
number of at a ae we and if an odd number be em- 
ployed, by +d, 3d, 3d, 
