HEYL & BRIGGS— THE EARTH INDUCTOR COMPASS. 23 



formly graduated dial, it is sufficient to replace the circle by a square, 

 as indicated in Fig. 7. Contact of the galvanometer leads is made at 

 points P and Q, whose distances from the center of the square vary 

 with the angle of setting. It may readily be seen that the resistance 

 included between -(- 6" and P will vary more rapidly per unit angle 

 of turn near the corner of the square than at the middle of a side. 



In Fig. 8, a is a constant and x a variable side of the triangle 

 having a constant angle 45° and a variable angle ^. In this triangle 

 the points P and -f 5' correspond to the similarly lettered points in 

 Fig. 7. In Fig. 8 we have: 



X a _ a 



sin (t> ~ sin (135° - (/>) sin (45° + 4>) 



a sin ^ 

 ^ = sin (45° + 4>) ' ^^^ 



Now, in Fig. 7 let R be the length of one side of the square; then, 

 if 4> be in the first quadrant : 



Potential at P = ( — 5 — ) sin — ( — ) cos d, 



Potential at Q = — { — ^ — I sin + ( — I cos > 

 If these are equal, 



X 



tan d = — 



R — X 



„ tan 6 „ sin 6 , . 



X = R = R • (7) 



I + tan sin + cos ' 



Substituting in (7) the value of x from (6) and noting that 



a= (R V2)/2, since R is the side of the square of which a is the 



semi-diagonal, we have : 



■\2 sin (f) _ sin , 



2 sin (45° + <^) ~ sin + cos ^ 



(8) expresses, for a square frame, the relation between the course- 

 angle 9 of the vessel and the null contact angle 4> of the dial switch- 

 board. (8) reduces as follows: 



^2 sin ■\2 sin 



2 sin (45° -\- (j)) ~ 2 sin 45° cos <^ + cos 45° sin 



sin <^ sin 6 



cos 4> -\- sin sin 6 -\- cos 6 



