Page:Scientific Memoirs, Vol. 2 (1841).djvu/209

 Rh than a given value $$W$$, It is clear that $$S$$ may either be one connected 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 enlarge or contract the space $$S$$.

Now let us assume $$P^{**}$$ to be a second point of similar properties to $$P^*$$ so that at it also $$V$$ may have a maximum value $$= V^{**}$$. As according to what has been before noticed, $$W$$ may have a value less than $$V^*$$, 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 $$V^{**}$$ shall not be less than $$V^*$$, it will be greater than $$W$$, and $$P^{**}$$ will necessarily 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 $$W$$ be 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 $$= V^{***}$$, characterised by being the lowest at which $$P^*$$ and $$P^{**}$$ are still situated in separate portions of $$S$$, which separate portions will unite whenever $$W$$ is diminished further.

If this union occur at a point $$P^{***}$$, the bounding line on which $$V = V^{***}$$ will have the form of an $$8$$, crossing at that point; where also we may easily satisfy ourselves that the horizontal 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 surface of the $$8$$, which involves a contradiction, 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 regards the points which lie outside. Figure 1. illustrates this form of the system of lines.