360 



Special Reports 



period variations in the correction-curves, not due to accident but possessing a charac- 

 teristic symmetry, had been either obliterated in the process of taking the means as 

 above outlined or rejected from the mean. To define more clearly these symmetrical 

 variations for the purpose of studying their character and discovering, if possible, their 

 cause, the method described in the following paragraphs was developed. 



Wherever one needle varies persistently from the mean of the other three, and 

 when this variation seems to follow a regular course as the inclination changes, or when 

 a needle gives a value at a single station which bears an unusual relation to the others 

 at that station, it is assumed that the mean of the three behaving normally is nearer the 

 true value of the inclination than the mean of all. The erratic needle then can be cor- 

 rected to the mean of the other three, and the value so corrected used for the further 

 study of possible variations of a similar character in the others. Suppose we have four 

 needles, Nos. 1, 2, 3, and 4, whose observed results at any station are a, b, c, and d. The 

 successive differences, (a b)=m, (bc)=n, (c d) = p, and, finally for check, (d a)=r, 

 are taken from the results at all the stations of a series arranged according to in- 

 clination and grouped so that there are, if possible, two to four group-values for each 

 degree of inclination. Not knowing in advance which needle requires correction, similar 

 differences are determined for all the needles, and trial-terms, 5, derived as follows: 



8 l = a-~(b+c+d) 



S 3 =c- j:(a + b+d) 

 Whence by substituting m, n, p, and r, 



Si 



5 (3m+2n+p) = (2m+n r) 



S 3 = | (3p+2r + m) = | (2p+r-n) 



8 s = b g(a-r-c-f-d) 



5i = d g (a + b+c) 



*= | ( 3n + 2 P + r ) = I (2n+p-m) 

 S4=|(3r+2w+) = |(2r4-m-p) 



(1) 



(2) 



The expressions (2) will usually reveal the needle showing the largest variations and 

 whether these variations are systematic. Assume that needle No. 1 shows such varia- 

 tions. Then since a appears in the expressions (1) for the first trial-term for each of 



needles Nos. 2, 3, and 4, each one must be modified by ^ Si giving the first error-terms, 



a, of the adjustment, as follows: 



<*i = Si 



2= 5 2 + o ^1 



3= ^3+ o ^1 



ai- 





(3) 



It frequently happens in the course of an expedition that more than one needle shows 

 these systematic variations, and these may overlap, covering regions of the same inclina- 

 tion. For discovering such, the observed values corrected for first error-term, and 

 designated a', b', c' , and d', are treated in a manner analogous to that outlined for the 

 original observations, and trial-terms, 8', for second error-terms in the adjustment are 

 derived thus: 



Si' = a'-~(b' + c' + d')=a-ai-^(b+c+d)+^(a i +a 3 + a i ) 



I 



b at g (a + C + d) + g (on + a 3 + cn) 



6,' = b'-~(a'+c'+d') 



c'-|(o'+6'4-d')=c-,-i(a+6+d) + |( l +ai+a) 



8.' 



h' = d'- J (a' + b'+c')=d-a t ~ i (a+b+c) + | (!+,+,) 



(4) 



