260 Theory of Transit Corrections. 



Then the observed times for a transit with five wires will be 



y =b d . sec <5±e 



y ± d . sec 8 -t e , 



y ± d . sec <5±e 3 



y qp d 3 . sec 3±e 3 

 7 =F d 4 . sec <5 zfc e 



and their sum, 5 y ± D. sec <J ± E = 5/ ± E since D=0. 



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Dividing by 5 we have y ± v ( 1 ). 



Now it is evident that if y is taken to represent the time of the 

 transit over the middle wire, and the equatorial intervals be equal, 

 the sum of the coefficients of sec d is zero, and there is a liability 

 to just as large an error of observation at each wire. Hence the 



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time of transit at the middle wire would be found y ± v (2) : a 



result no more accurate for the middle wire than was the pre- 

 vious one for the mean of the wires. 



If to expression (1), the product of the equatorial distance of 

 the mean of the wires from the middle wire be applied with the 

 proper sign, y, which represents the true time of passage over the 

 mean of the wires, will become the true time of passage over the 

 middle wire, and the expression (1) will become identical with 

 expression (2) : the same result will be obtained as if the equato- 

 rial intervals had all been equal. 



The correction for the inequality of the intervals may also be 



made, by reducing the times of the transits on one side of the 



middle wire, to what they would have been, if each wire had 



been as far from the middle as is the one corresponding to it 



the other side, and then using these reducing times in obtaining 

 a mean. 



Let y - true time of passage at the middle wire d', d", etc. = 

 true times between the middle and nthm- vmi 

 Then y ±-d 



on 



Y =fc a' ±e 



Y =h«J" 



Y =r= d>" ± e 3 



Y^d 



will be the observed times of the transits. Now conceive a quan- 

 tity ± b added to d", such that d" ± b = d'", and also another 

 quantity ± b' added to d', such that d'±b'=d"". In this case, 

 when the quantities are added, the second column will disappear, 



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and dividing by 5, we have y± t : as accurate a result as is ob- 



tained when the intervals between the wires are all equal ; since 

 the observer is liable to just the same errors at the wires. 



