198 



C. F, GAUSS ON THE GENERAL THEORY OF 



If the junction take place at two different points, what has 

 been demonstrated for one point would hold good for the two ; 

 and one may easily see that inside the space inclosing P* and 

 P** an insular space would be formed, which would gradually 

 contract itself as W was diminished, and w^ould necessarily at 

 length resolve itself into a true south pole. 



The case is similar when the junction takes place at three 

 or more separate points ; but if it take place at once on a whole 

 line, then the horizontal force must disappear on all the points 



of that line. 



It is evident that the assumption of two south poles would m 

 like manner necessitate the existence of a third polar point, 

 which would be neither a south pole nor a north pole, or rather 

 would be both at once. 



13. 



From what has been developed in the foregoing article, its ap- 

 plication to many conceivable exceptions from the simplest type 

 of our system of lines will be readily understood. The whole 

 of the points to which a certain value of T corresponds, may 

 be a hne consisting of several portions, of which each retiu-ns 

 back into itself, but which are quite separate from each other ; 

 it maybe aline crossing itself ; lastly, it maybe a line having on 

 both sides spat es where V is greater than on the line, or where 

 it is less. 



We may assert that on the earth there are, on the great scale, 

 no deviations of such a nature from the simplest type. 



