70 DONALD C. BARTON 
traverse is comparable with the difference, (d), between the duplicate 
measurements of the length of a line with the surveyor’s chain; and 
the number of torsion balance stations, V, is comparable with the 
number of applications of the surveyor’s chain. 
The formula (6), therefore, may be re-written in the notation 
which we have assumed for the torsion balance surveys, to read: 



PE = 0.6 V/ sl a apy 
= Oo. ee 
(7) DOCU SANG 5 i Te N. 
The error of closure, EC, is in terms of Ag (that is, cm. sec.~*) and the 
probable error (PE), therefore, would be given in terms of Ag per 
unit of measurement which according to our preceding assumptions 
is the station interval. The latter rather commonly is fairly constant 
within a traverse, but is variable from traverse to traverse. The cal- 
culation of Ag over a station interval has the mathematical form of 
a gradient times distance, that is, times the length of the station 
interval; and, therefore, any error in Ag over a station interval will 
tend to be lineally proportional to the length of the station interval. 
The calculated “probable error” in terms of Ag and of the inconstant 
station interval would be useless. The ‘“‘probable error,’’ however, 
may be reduced to the dimensions of a gradient and then will be in- 
dependent of the length of the station interval. All error in the cal- 
culation of Ag over a station interval may be assumed to be in the 
gradient, if, as we usually assume, the stations are close enough to- 
gether so that the variation of the gradient is linear from station to 
station. The probable error in Ag over a station interval, then, will 
vary linearly with the length of a station interval and may be ex- 
pressed as the product of the length of the station interval times that 
probable error in the gradient. Conversely, the probable error of the 
‘gradient may be expressed as the quotient of the probable error in 
terms of Ag over the station interval divided by the length of the 
station interval. If the terms 
ee; 
N 
of formula (7) are weighted in terms of the square of the reciprocal 
of the length of the station interval, SJ, to read 
EC?) ee ee 
VSI)? Neste NoSTe 


400 
