INTENSITY OF THE EARTH'S MAGNETIC FORCE. 263 



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 

 0, 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 inclination. 

 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, rj 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 pro- 

 bable errors of TJ and /, due to this cause, vary with the angles 

 themselves. To determine their magnitude in any case, we have 



mR sin (0 - TJ) = F, 



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

 taken into account, 



mR sin (0 - rj + Arj) = F + f; 



f denoting the moment of friction, and rj - Arj the new angle of 

 equilibrium. Developing the latter equation, and subtracting the 

 former, 



mR cos (0 - rj) AJ/ = ./' ; 



the angle A) being expressed in parts of radius. Hence, cos (0 - ri) A j 

 is constant with a given instrument, and at a given point of the 

 Earth's surface. 



To find the probable error of the force corresponding to the 

 error of the observed angle, we must differentiate the equation 

 of equilibrium, mR sin u = F, with respect to R and w, where 

 u = - r j ; and we have 



A R sin u + R cos u A = 0. 



But 



i / \ 



u 5 I 1 ? 1 ~ 1v 



tj, and rj, being the observed angles of inclination under ill 

 opposite actions of the deflecting force. Hence the probable error 



of u is 



- 1 



Au = i -y/Arj'i + ATJ".. = , ' ' 



v/x, 



