INTENSITY OF THE EARTH'S MAGNETIC FORCE. 249 



whence we obtain the following equation for the determination 

 of q : 



3^0 _ 5 ? 8 _ 2 = o. 



In order to solve this equation, we may observe that, q being 

 greater than unity, the last term of the equation may, in a first 

 approximation, be neglected in comparison with the others; so 

 that we have, approximately, 



And setting out from this value, we find, by any of the known 

 methods of approximation, 



q = 1-32 ; 



or 1|, very nearly. Accordingly, the smaller distance, r, being 

 determined by the condition that the third term of the series shall 

 be insensible, the greater distance should be T32r. 



If we substitute this value of q, in the expression for -^- above 

 obtained, we find 



from which we can calculate the least probable error corresponding 

 to any given angle of deflection, the probable error of reading 

 being known.* 



Now let us suppose the term containing the fifth power of the 

 distance to vanish, in the expression for the deflecting force. The 

 value of Q will then be reduced to 



Q = D 3 tan u ; 



* In the Dublin Magnetical Observatory tbe deflecting bar hitherto employed is 

 12 inches in length, and the least deflecting distance therefore 4 feet. The deflection 

 produced by it at this distance is about 3 56' ; and the probable error of observation 

 does not exceed 5". Hence, in this case, 



AQ 5-563 



Q ~ 236 x 12 

 The absolute intensity, X, varies inversely as the square root of Q ; so that 



Consequently, the resulting probable error in the determination of the absolute 

 intensity, made according to the usual method, is, at this Observatory, about the 

 TuVoth part of the entire quantity. 



