92 G. H. KNIBBS. THE THEORY OF THE 



if g — T' and_/; = l" twenty repetitions will give a result with a 

 probable error of + 0'"59 ; if g were wrongly estimated and in- 

 cluded as 5", it would appear as 0."48, but if it were neglected 

 entirely only 0. "32, little more than half its proper value. 



Algorithm of the process of evaluating errors of pointing and of 



reading. 



If in (13) each quantity be subtracted from the next following, 

 the result is the following succession of values for the angle 

 measured, and hence this process exhibits symbolically the errors 

 of the several measures, regarded as independent observations of 

 the angle. Each measure is affected by two errors of pointing 

 and two of reading, as the following series shews : — 

 « -pi +p» -9i +9-2, a-p s +Pi-g 2 +ff 3 a-P2p.-i+P2a.-gn 



+ ffn+U I 1 ")- 



When the arithmetic mean is subtracted from each of these 

 quantities, which represent the several measures of an angle as 

 shewn in I. hereafter, the residuals, (remainders, or differences) 

 are individually affected by errors resulting from the combination 

 of the two errors of reading and two of pointing, and the estima- 

 tion of the probable error of a single measure in the ordinary way 

 gives (approximately) the probable error of one measure of an 

 angle subject to the errors defined. 



Using v as the symbol to denote a residual 2i> and 2v 2 for their 

 sum and the sum of their squares, the probable bi-elemental error of 



one measure is, somewhat approximately, -675 »/ ■"- — -...(18) or 



I 2v \ n ~ . 



•845 V < — — > (19), the latter formula being applicable 



rather when the number of observations is large, than generally. 

 It is not legitimate however to determine the probable error of 

 the mean of n such measures by dividing (18) or (12) by iv/n, as in 

 the case of independent and collateral measures, and for this 

 reason, viz, that although each measure per se is subject to two 

 errors of pointing and two of reading, the peculiarity of the method 

 of repeating is that, while leaving the errors of pointing unaffected, 

 it eliminates every error of reading (as the terms in (13) clearly 

 exhibit) except the primal and final. As a consequence neither 

 the probable error, nor the weight of the mean result, varies 

 (respectively) reciprocally or directly as the number, or as the 

 square root of the number of measures, nor does either admit of 

 any antecedent general evaluation.* 



Reverting to (18) and (19) the error of the arithmetic mean of 



the measures in (17) is evidently +ltzl±±£*+i (20), 



n 



* Except in the restricted form exhibited in Table (III.) hereafter. 



