124 DR L. BECKER ON THE SOLAR SPECTRUM. 



two positions of the grating. Therefore any personal error in bringing the lines to the 



cross wires would be eliminated if it depended on the direction of the motion. 



In order to form an idea of the working of the apparatus 142 lines with 1583 single 



observations were selected, which lay about half-way between standard lines, and were 



distributed over the whole length of the spectrum. They gave as the probable error of a 



single observation 



r= ±0056 A.U. 



» 



This value corresponds to ^-th inch on the recording paper, to YTT* n mcn on the circumfer- 

 ence of the first wheel, and to ^,ViRyth inch on the circumference of the worm-wheel. Every 

 line having been observed on an average eleven times, the probable error of the wave- 

 length of any well-observed line lying half-way between two standard lines amounts to 



r= ±0-019 A.U. 



It would have occupied too much time to repeat the same computation for all the lines. 

 We therefore chose an entirely different way. In the course of the computations we had 

 deduced the means of the wave-lengths for every line, as well from the high sun 

 observations, as from those of the low sun in both positions of the grating. There are 

 thus three series of results belonging to the same lines. 



Let s ± and s 2 designate the values given in two sets of results, and m the true value. 

 The average error n of one value is then : 



„ ■ [si-m] . _ . fa-ro] 



if n is the number of values in each set, and [ ] stands for the sum irrespective of the 



signs. 



Let s x be the mean of p, and s 2 of q observations, and suppose all the observations 



equally accurate. 



fo-m] J'p - [s 2 -m] Jq = 0. 

 Further 



[Sj-m] + [s 2 -m] = fo-sj 



when the true value m is supposed to lie between s x and s 2 . 



This granted, the average error of one value resting on p + q observations becomes : 



Jp+q(Jp+Jq) n 



The factor being symmetrical with respect to p and q, the same formula will hold good 

 if p and q be interchanged for any pair of values. 

 The probable error follows by the known relation 



r = 0-845??. 



In using this formula we are well aware that neither the condition of equal accuracy, 

 nor that of equality in the number of observations, is strictly fulfilled. Comparing the 



