Dip-Needle Errors 



361 



These equations by use of (1), (3), and the condition (51 + 52 + 53 + 54) =0, become: 



5/ = 



5 2 ' = 



1 



<*2 



03 ~ o 3 



04 o a * 



(5) 



Supposing that needle No. 4 shows variations from the mean which need further cor- 

 rection, the second error-terms are deduced thus: 



<*1 5j + q 54 



2 5 2 + q 54 



3 



5j + q 5 4 



' _ s ' 

 <*4 = 04 



(6) 



If there are more than two of the four needles in need of correction within any given 

 region, the problem is indeterminate, though where the variations overlap but slightly 

 their presence may be detected by a graphical process. Where there are two such erratic 

 needles, doubt often arises as to which should be corrected on the first trial, and it is 

 evident that the absolute value of the resulting error-terms will vary with the magnitude 



of the 5 chosen since each preliminary trial-term is increased by 5 5. The final corrections 



of the four needles will retain the same relation independent of the choice made. In 

 such cases the choice must depend on the relations between the needles in contiguous 

 regions, where inclinations vary but slightly, and where the observations were made at 

 about the same time. This process of analysis may be quite easily and rapidly performed 

 by arranging the work in suitable tables, each step in the process furnishing its own check ; 

 the differences between the resulting corrections should agree very closely with the needle- 

 differences of the groups from which they were derived. Assuming that the mean of all 

 needles in a given instrument requires a constant correction, A, independent of the 

 value of the inclination (the assumption upon which the present method is based), the 

 correction Aln for any needle, n, becomes 



A/ n = A-(an + a' n ) (7) 



To illustrate the features of short-period corrections of the character under consider- 

 ation, several examples are presented. The first is taken from the work of Observer 

 F. Brown during March to July 1915 in eastern China with dip circle No. 177 using 

 needles Nos. 1, 2, 5, and 6. In this case the stations are not as closely arranged as is 

 desirable, but the values are sufficiently well verified to warrant their use. The correc- 

 tion-curves as actually applied to these observations (see p. 14) were smoothed some- 

 what more than the ones shown in Figure 4, in which the individual points as they were 



Fio. 4. Correction-Curves for Needles used with Circle No. 177 during March to July 1915. 



