344 MAGNETIC METHOD [Chap. 8 



Substituting 90 — I — 95 for t; in eq. (8-16), the magnetic couple 

 for a = is 



Di = MH sin (90 - I - <^) + MZ cos (90 - I - <p). (8-37a) 



When Di = 0, 



MH cos I + MZ sin I ,_ ^_, . 



tan if = ■■-„ . T 7iT>r T- (8-376) 



MB. sin I - MZ cos I 



Substituting H/T for cos I and Z/T for sin I, tan ^ = 00 and (p = 90°. 

 Without the counterweight the needle points in the direction of the in- 

 clination. For intensity determinations the gravity action due to the 

 counterarm must be considered. It is proportional to the mass ni', its 

 arm e, and the angle \p; hence, the moment D2 = —em'g cos 1/'. Sub- 

 stituting rp-\-^ = 90-l-<pSind^ = xP-\-<P (from Fig. 8-28), 



2)2 - -em'g cos (^ - <^). (8-37c) 



In the position of equilibrium, Di -\- D- — 0, or 



MH cos I -f MZ sin I — em'g cos ^ .^ „_ ,. 



MH sm I — MZ cos I + emg sm ^ 



Substituting cos I = H/T, sin I = Z/T, and (H' + Z')/T = T, 



Ml - em'g sin (I -}- f) . , 



tan ip = , 7 t I ys — • (8-37e) 



emg cos (I + ^) 



For a critical angle fo = 90° — /, v? becomes 90° and the system swings 



into the direction of the inclination. For two total intensities, T' and T 



M(T' — T) 



the two corresponding deflections are given by tan <p' — tan <p = -. — -. — — , 



emgsm^ 



which for small angles, is A(p — eAT, where the scale value 



^^ em'g^n^ ^ (8-37f) 



Note the similarity of this scale value and that of the vertical Schmidt 

 balance, (e = mgd/2fM). The e sin ^ corresponds to (^ = e sin f. A 

 diagram showing the sensitivity as a function of angle ^ and further details 

 concerning the operation of the instrument are given in Steam's article. 



6. Early Swedish and American prospecting instruments. Of simple 

 construction, these instruments were designed for the rapid measurement 

 of large magnetic anomalies and are still in use for rapid reconnaissance 

 in many mining districts; the use of more accurate instruments there would 

 offer no particular advantage. 



23 Ibid. 



