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

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.