TERRESTRIAL MAGNETISM. 197 



than a given value W, It is clear that S may either be one con- 

 nected surface or several detached spaces, and that V = W, on 

 the bounding lines or lines which separate S from other parts 

 where V is less than W; by increasing or diminishing W, we en- 

 large or contract the space S. 



Now let us assume P** to be a second point of similar pro- 

 perties to P* so that at it also V may have a maximum value = 

 F**. As according to what has been before noticed, W^may 

 have a value less than F*, and differing from it by so small an 

 amount that P** shall fall outside that part of S in which P* is 

 situated ; then if we arrange (as we may do) that F** shall not 

 be less than F"*, it will be greater than W, and P** will neces- 

 sarily also belong to a part of S. Thus P* and P** will both 

 be situated in S, but in separate portions of it. On the other 

 hand, it is evident that W may be taken so small that P* and 

 P** shall both be situated in one connected part of S ; for by 

 only taking W small enough, S may be made to embrace the 

 whole surface of the earth. 



If then Whe made to pass progressively through all the values 

 between the first and the second values spoken of, there must be 

 amongst them one which we will call = F***, characterised by 

 being the lowest at which P* and P** are still situated in sepa- 

 rate portions of S, which separate portions will unite whenever 

 ^is diminished further. 



If this union occur at a point P***, the bounding line on 

 which V = F*** will have the form of an 8, crossing at that 

 point ; where also we may easily satisfy ourselves that the hori- 

 zontal intensity must = 0. In fact, the crossing either does or 

 does not take place under an angle of sensible amount. 



In the first case, the horizontal force, if it be not = 0, must be 

 directed in the normal to the two different tangents, which is 

 absurd; in the second case, in which the two halves of the 8 

 touch each other at P***, or would have the same tangent, the 

 force normal to this tangent could only be directed towards the 

 interior of one half sui-face of the 8, which involves a contradic- 

 tion, as the value of V increases towards both sides ; therefore 

 P*** is a true magnetic pole according to our definition, but 

 must be considered as a south pole as regards the points nearest 

 to it inside the two openings of the 8, and as a north pole as re- 

 gards the points which lie outside. Figure 1. illustrates this 

 form of the system of lines. 



