30 HEYL & BRIGGS— THE EARTH INDUCTOR COMPASS. 



If the field be inclined at an angle 6 to the horizontal, as in Fig. 

 12, we may resolve it into two components, H sin 9 vertically and 

 H cos horizontally. If the intensity of the currents in each half of 

 the ring in Fig. ii be taken as unity, the intensities of the similar 

 currents generated in Fig. 12 will be sin and cos 9, respectively. 

 These currents are taken off by two pairs of brushes, and encounter 

 the same external resistance in the dial switchboard. We may con- 

 sider these component currents as superposed, the intensities in the 

 four quadrants being shown in Fig. 13. 



Let the radius of the ring be unity and let L be the inductance 

 per unit arc of the ring. Then, representing the current intensity at 

 any angle ij/ (Fig. 14) by i, the total cross-field C will be propor- 

 tional to 



2L I icos (9— \j/)d\l/ 

 Jo 



where 9 — i/' is the angle between the reaction-field of the element of 

 the ring at if/ and the perpendicular C to the field H. 



Using the values of * in Fig. 13 this integral breaks up into two. 



2L (sin <? -{- cos 0) M cos (9 - xp)dxP 

 Jo 



-f- 2L (sin 9 - cos 9) \ cos {9 - \P)dxl/ 



_ 



2 



= 2L (sin 9 + cos 9^ + (sin 9 - cos 9)'- 



= 4L. 



Hence, the resultant cross-field reaction of the ring is independent 

 of the course-angle 9, and there is no quadrantal or other segmental 

 error from this source. 



The instrument as above described will give about one millimeter 

 ■deflection at the galvanometer for five degrees change of course, the 

 sensitivity being a little greater at the cardinal points than in the 

 middle of a quadrant. This sensitivity is capable of a reasonable 

 amount of increase, if desired, by using a larger armature with a 

 greater number of turns, or by increasing the speed of rotation, but 

 for the present state of the art of aerial navigation the instrument is 

 sufficiently sensitive. 



