538 The Eev. Dr. Lloyd on the Determination of 



2. When the planes in which the needles move are perpendicular to the 

 magnetic meridian, or a= 90°, a' = 90°, the left-hand members of (1) and (2) 

 become, respectively, MY cos f}, Yco& rf ; whence 



F" cos ij cos ri = UWr. (4) 



3. Finally, the equilibrium may be produced, in both cases, by turning the 



instrument in azimuth until the free needle stands vertically. In this case 



»; = 90°, i/ = 90', and the left-hand members become —MX cos a, —Xcosa'; 



whence 



X^ cos a cos a' = UWr. ( 5 ) 



Thus we may apply this principle to the determination of the total inten- 

 sity, or to that of either of its two components. 



In comparing the foregoing methods, it is to be observed that the third fails 

 when the inclination approaches to 90°, on account of the magnitude of the 

 error of R resulting from a given error of 6, when the total force is deduced 

 from its horizontal component. In like manner, and for the same reason, the 

 second method fails in the vicinity of the magnetic equator, or line of no in- 

 clination. The first alone is applicable at all parts of the Earth's surface, 

 and I proceed to consider it more in detail. 



The observed angles, j; and )/, are liable to error, the friction of the needles 



on their supports causing them to rest in positions slightly different from those 



due to the acting forces. The probable errors of i/ and i/, due to this cause, 



vary with the angles themselves. To determine their magnitude in any case, 



we have 



MR sin (e - ij) = F, 



F being the moment of the deflecting force ; and when friction is taken into 

 account, 



MRsm {e - ); -^ Ay)=F+/; 



/ denoting the moment of friction, and y—A}] the new angle of equilibrium. 

 Developing the latter equation, and subtracting the former, 



3IR cos (e - y) Ai] =f ■ 

 the angle A); being expressed in parts of radius. Hence, cos (6 — ??) Ar/ is con- 

 stant with a given instrument, and at a given point of the Earth's surface. 



