GAUSS'S THEORY OF TERRESTRIAL MAGNETISM. 117 



49. Continuation of Gauss's investigation : applica- 

 tion in a numerical form. 



Put p for sin u . sin u . cos (X X ) + cos u . cos w , and 

 expand the fraction for - . In the paucity of well-de- 



termined elements, and in the complexity of expressions, 

 Gauss thought it sufficient to develope this to the 4th 



power of r . This gives for -, 



S -"'+"/) 



= 

 r 



cosw.cosX cosw.sinX. ft 



~i - ( r cosw .smX ) 



sin u 



The corresponding term of V will be 



cos u. cos X fcx 

 --- 2 -- I O/A . r . cos w . cos X 



. sinw 



Each of these integrals is an unknown constant. Call- 

 ing them i v i y i# the term of V will be 



cos u . cos X . cos u . sin X . sin u . 



