372 Astronomical and Nautical Collections. 



ill. Computations for clearing the Compass of the Regular Effect 

 of a Ship's Permanent Attraction. 



Investigation. 



1. A SUFFICIENT approximation, for the explanation of many 

 of the phenomena of the dipping needle, is obtained by sup- 

 posing the magnetism of the earth to be concentrated into two 

 magnetic poles, very near to each other, and to the earth's 

 centre ; this supposition being also equivalent to that of an 

 infinite number of small magnets, parallel to each other, distri- 

 buted equally throughout the earth's surface, or through any 

 other concentric strata. 



2. The angular distance of any point on the earth's surface, 

 from the equator belonging to these poles, being called the 

 magnetic latitude, it has been demonstrated by several mathe- 

 maticians, that the tangent of the dip must be twice the tangent 

 of the magnetic latitude. 



.3. Hence it may be inferred, that if the sine of the dip be 



called s, that of the magnetic latitude will be ,-- — - — ,. 



v/ (4 — 3ss) 



4. The angle subtended at any point by the two poles will 

 obviously vary as the cosine of the magnetic latitude. 



5. Consequently, in the triangle representing the two mag- 

 netic forces and their result, either of the two greater angles 

 being uliimately equal to the complement of the dip, it follows 

 that as the cosine of the dip is to the earth's radius, so is the 

 sine of the small angle, subtended by the two poles, to the side 

 corresponding to the ultimate magnetic force in the direction 

 of the dipping needle. 



6. The magnetic force in the direction of the dipping needle 

 will therefore vary as the cosine of the magnetic latitude di- 

 rectly, and inversely as the cosine of the dip, or as — ^r ; or 



■' •' cos. D 



^ sin. L sin. L ta. D ta. D sin. L „ sin. L 



since cos. L= - — —, as-——. - — 7: =- — —. -. — ^=2 -: — - 



ta. L ta. L sm. D ta. L sm. D sin. D 



2 



— — - ; and the magnetic force must vary inversely, as 



-^(4_3ss) * J J' 



the square root of 4 diminished by three times the square of 



