Dip-Needle Errors 369 



Rust patches will, of course, take on an indefinite variety of irregular forms, among 

 which there may be such as would cause the short-period variations under discussion. 

 Suppose a rust mass, such that a cross-section through it at the point of support will have 

 a circular outline with radius r', the center of curvature being at C", as shown in Figures 

 9a and 96, in which C is the center of the axle, LTP and L'T'P are the common tangents 

 to the two surfaces, CL and CL' are the radii drawn to the points of tangency on the 

 axle, the angle between which is 28. If CI represents the direction of the field when the 

 needle is in equilibrium at rest upon the point G and the direction of the field is changed 

 by a small increment /3, then the needle will roll forward under the influence of gravity 

 through a small angle e until a new equilibrium is established. This will occur when 



wg(r+kr') sin (0 + t) = Fm sin e (13) 



where h represents the thickness, HG, of the rust patch at its center; h may be dis- 

 regarded without appreciable error, as it is very small in comparison with the radius of 

 the axle. Since HP = r (sec 8 1) and GP = r' (sec 8 1) it follows that 



h = (sec 8-1) (r-r') (14) 



Thus for 8 = 2 the thickness of the rust patch at its center would be only 0.0006 (r r'). 



Hence from (13), placing ^ ' = A', 



m 



^F^P (15) 



Since for maximum value of t, /3-f-e = 0, 



A' = e maI - F (16) 



8 



An application of these formula? may be made to data taken from the curve for 

 needle No. 5 between inclinations +61 and +64. 5 (see Fig. 6). For this case /?, the 

 angle of inclination-change between the center of symmetry and the maximum values of 

 A/n, is approximately 60', and 2d, the range in inclination between the beginning and end- 

 ing of the short-period variation, is approximately 3. 6; hence from equation (11) 

 ,,. is 48' approximately. The average value of F over the range of inclination involved 

 is 0.55 c. g. s. From these data and equation (16) we have A' = 0.244, and from equation 

 (15) computed corrections, A/ B = 0.100/3 between maximum and minimum, and 

 A/ 6 = 0.125 (0-/3) beyond the maximum and minimum, as seen by equation (11) 

 and shown in Figure 10. The adjusted corrections from observations, each from a group 

 of separate stations, are indicated in the figure and show a substantial agreement with 

 the values computed on the above basis. The dimensions of a rust mass that would 

 produce the short-period variation as above may be computed approximately. Since 

 A (r r') =A'r and taking A for needle No. 5 as 0.474' and r as 0.025 cm. the value of 

 r' for ^' = 0.244 is 0.012 cm., whence, from (14) for e = V.8 as above, h = 0.000006 cm. 

 approximately. The diameter of the rust mass on the pivot would be about 0.002 cm. 



The behavior of needle No. 1 in circle No. 177 indicated that conditions causing 

 changes in correction-curves may arise suddenly during field work. There is also evi- 

 dence that the short-period variations are not necessarily permanent. Evidence of this 

 is found in the behavior of needle No. 5 in circle No. 177 which showed large short-period 

 variations between inclination +61 and +65 during the expedition in Mongolia 

 (Fig. 6). At the conclusion of that expedition a shorter trip was made by Observer 



1 Needle No. 5 was broken in the field and was not available for the determination of .4 ; the value determined for it* 

 mate, No. 6, is therefore adopted. 



