FOR DETERMINING THE MEAN DENSITY OF THE EARTH. 
325 
different results, for we should thus neglect all comparisons except the first and 
the last. 
40. Let the comparisons be numbered 
0, 1, 2, 3, 4, n—\, n, 
and let the swings be numbered 
1, 2, 3, 4, n. 
Let the errors of comparison, estimated by their effect on a four-hours’ rate, in 
units of the last figure of 8-figure logarithms, be 
Eo5 Ei, Eg, Eg, E 4 , E„_i, E„, 
(where E,g is supposed to be taken as if the comparison, which was really omitted, 
agreed with that produced by interpolation between the two neighbouring compari- 
sons,) and let the probable errors of comparison be 
^15 ^25 ^3 3 ^4 3 ^»-13 ^«3 
and let the weights for the results of the separate Swings be 
^13 ^23 ^33 ^45 • ^ n ’ 
Then the errors in the results of the separate Swings, produced by the errors of 
comparisons, are 
(E, — Eg), (E 2 — El), (E 3 — E 2 ), (E„ — E„_i) ; 
and, combining these with the weights 
^c ,3 w .,, 
the actual error of the final result will be 
^l(Ei — — Ei)+W 3(E3 — Eg)-f- -\-Wn{^n — E„_i) 
W, +W^ +W^ + +Wn 
{w■^ — W^'Fn + — + — +Wn^n 
W] +W^ +W3 -f +Wn 
Hence, by the well-known rules of the Calculus of Probabilities, the square of the 
probable error of the final result will be 
K’^0+K-^2)"-^H(^2-^^3)'^l+K-^4)'^l+ +<‘<_ 
{Wi + W 2 + W3 + +WnY 
And if, for simplicity, we suppose all the comparisons equally good, so that for e^, Cj, 
63, &c. we may put e, the expression for the square of the probable error of the final 
result becomes 
g2^.^ ^HK-^2)'+(^2-^3)'+ +< . 
{Wi -f W2 +.W3 + 
and we have now to determine values of tc,, &c., which will make this square of 
the final probable error a minimum. 
