17 



and if this équation is not satisfied, the cause must be an accumulation of 

 errors of observation. In order to satisfy it rigorously, \ve shall correct F„ 

 and (t„, and adopt as our corrected values: 



F = F 



G., 

 ~ (9) 



so that the condition: 



G',. = G„ - "2 (10) 



F'„ + r?'„ = 



will be rigorously satisfied. It is necessary to considcr for a moment the 

 possible sources of error which cause the quantity 



F,,-VG,. 



to differ from zéro. We must bear in mind that the intercomparison of diffé- 

 rent lines on the reseau cannot be made directly, but must be effected by 

 comparing each liuc separately with the scale of the Repsold machine. To 

 fix our ideas, let us imagine that in a given instance the sum: 



F„ + G„ 



has a definite positive value Q. This quantity Q is then the sum of the ex- 

 cesses of lengths measured on the end lines of the réseau, över the correspon- 

 ding length on the middle line. Clearly Q may arise because we have made 

 the middle line too short in comparing it with the scale of the machine, or 

 because we have made the end lines too long. The one thing is as likely to 

 have occurred as the other. Therefore it is best to distribute Q equally among 

 the comparisons with the scale of the three lines. Evidently we shall reduce 

 Q to zéro, if we lengthen the middle line by \ Q, and shorten both end lines 

 by the same amount. The excess of each end line över the middle will then 

 be reduced by i Q, and the sum of the excesses will be reduced by Q, thus 

 bringing it down to zéro. Accordingly we have assumed that any distance 

 taken along the middle line is really longer than it was found to be upon 

 comparison with the scale of the machine by the quantity: 



