HEYL & BRIGGS— THE EARTH INDUCTOR COMPASS. 21 



lively). Fig. 4 represents the distribution of potential in this case. 

 The potentials at -|- 5 and -{- C are no longer equal, that at -]- 6* being 

 greater than that at -|- C. In consequence, the line of contact will be 

 shifted, as it should be, a little nearer to the horizontal than in Fig. 3. 



It is a curious fact that in general the angle of the null contact line 

 with the ± 5" line is not always equal to the course-angle 6 of the 

 vessel with the magnetic meridian. There is, as we have just seen, 

 perfect correspondence at the cardinal and 45° points; but for points 

 within each octant, as in Fig. 4, there is a departure rising to a 

 maximum of about 4° near, but not at, the center of each octant, as 

 the general mathematical theory will show. 



Assume the 5" and C voltages applied as in Fig. 5 (same as Fig. i) 

 and let the contact line of the galvanometer be inclined at an angle (f> 

 to the ± S line, measured positively in the clockwise direction. We 

 shall first suppose </> limited to the first quadrant. 



{90° - 4>\ f "!> \ 



Potential at P = ( 5 — ) sm — I —3 I cos 9, 



\ 90 / \90 / 



/go" - (f>\ . f <f> \ 



Potential at Q = — I ' o — I sm ^ + | — 5 I cos 6. 



\ 90 / \90 / 



If these are equal, we must have 



Since equation (i) is algebraic in (f> and transcendental in 6, it is 

 evident that no linear relation can exist between cf> and 9. Solving 

 for </> we have 



T tan 6 T sin 6 , , 



2 I + tan ^ 2 sm + cos ^ ^ ' 



Now 



■K sin Q . 



*^~^=2s"In'0 + cos^~^' ^^^ 



which will be a maximurn or minimum where 

 d{<^ - d) T I 



de 2 (sin d + cos 6)^ 



-1 = 0. (4) 



