TERRESTRIAL MAGNETISM. 193 



perly speaking, are only different modes of expressing the same 

 thing) may be tested, at least approximately, by a refei-ence to 

 observation. 



Let P°, P', P" .... P" be a polygon on the surface of the 

 earth, the sides of which are the shortest lines that can be di*awn 

 between their respective extremities, and are therefore portions 

 of great circles, the earth being here considered simply as a 

 sphere. Let ocP, co', co", &c. be the intensities of the horizontal 

 magnetic force at the points P^, P', P", &c. ; further, let 8", S', 

 B", &c. be the decUnations reckoned in the usual manner, west 

 of north as positive, east of north as negative; lastly, let (01) 

 be the azimuth of the line P'^ P' at P°, reckoned in the customary 

 manner, from the south by the west ; in like manner (10) the 

 azimuth of the same line taken backwards at P', and so on. 



Let it be observed that t alters continuously in each of the 

 sides of the polygon, but suddenly at the corners, where there- 

 fore ithas two different values ; for example, at P, t has the value 

 (10) + B', in consideration that P' is the end of the line P" P' ; 

 and the value of 18^ + (12) + S', in regard that it is the begin- 

 ning of P' P". 



We may consider the approximate value of the integral 



/ o) cos t .ds, extended through P° P', to be 



i {aP cos f + w cos V) . P'^P', 

 where t^ and t' signify the values of t at P° as the beginning, 

 and at P' as the end of P° P'. This approximation is all that 

 can be obtained, because we have the values of co and / only at 

 the extremities P^ P', and is deserving of confidence in propor- 

 tion to the shortness of the line. The given expression is, in our 

 notation, 



= i(ft)'cos ((10)) + B') - ft)Ocos ((01)) + 8°]) .P^P'. 



In like manner, the approximate value of the integral, extended 

 through P' P", is 



= i(a>' cos ((21) + 8") - a>'cos ((12) + B']).P'P", 

 and so on through the whole polygon. 



Therefore, for a triangle our proposition gives the approxima- 

 tively correct equation 



o)" {P^P' cos ((01) + 8°) - P^P" cos ((02) + 8"^]) 

 + ft)' (P'P" cos ((12) + 8') -pop' cos ((10) + 8']) 

 + a»" (Po P" cos ((20) + 8") - P' P" cos ((21) + 8"]) = 0. 



