FOR DETERMINING THE MEAN DENSITY OF THE EARTH. 
327 
nience, make h=\. Then the weights for the results of the successive Swings are, 
n, 2n — 2, 3w — 6, 4w— 12, &c. 
N 
The square of the probable error of the final result was found =e*x^2- Substi- 
12 
tuting, this becomes e^X — ^ ; 
° n.n + l.n + 2 
or the probable error =ex 
12 
n .n-\- I . re + 2 
41. It will be instructive to contrast this result with the result obtained on two 
other suppositions. 
First: suppose that the Swings had been continuous, but that there had been no 
intermediate comparisons of clocks. The probable error of the first comparison 
being e, and that of the last comparison being also e, the probable error in their 
combination by subtraction will be es/'2 ; and as this applies to n Swings, the pro- 
bable error on the mean =^e\/2=ey/^. Comparing this with the probable error 
found above, it appears that the intermediate comparisons have diminished the pro- 
bable error in the proportion expressed by the fraction Wh en w = 26. 
this fraction is 
or the weight of the result is increased nearly five-fold by the 
intermediate comparisons. When w=15, the fraction is \/ ot- the weight is 
increased three-fold. 
Second : suppose that the Swings had been discontinuous. The probable error in 
each Swing, found by combining its first and last comparison, would have been es/2 ; 
and, as the different Swings are strictly independent, the probable error on the mean 
of all would have been Comparing this with our probable error above, it 
appears that our system has diminished the probable error in the proportion 
\/ ^ When w = 26, this fraction is a / or the weight of the result is 
increased 126-fold by our system. When w=15, the fraction is weight 
in increased 45-fold. 
These contrasts will suffice to show the great advantage of a system of continuous 
Swings with intermediate comparisons such as has been employed in this experiment. 
I cannot quit this subject without repeating that my first impression on the advan- 
tage of such a system was derived from the representations of Mr. Sheepshanks*, 
on occasion of the experiments of 1828. 
42. In the First and Second Series, w=26, and the successiv^e weights are 26, 50, 
72, 92, no, 126, 140, 152, 162, 170, 176, 180, 182, 182, 180, 50, 26. In the 
Third and Fourth Series, w=15, and the successive weights are 15, 28, 39, 48, 55, 
* Since I commenced drawing up this paper, my valued friend has been snatched away by death ; a victim, 
1 believe, to his labours gratuitously undertaken for the formation of the National Standard of Length. 
