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IX. Geometrical Investigations concerning the Phenomena of Terrestrial Magnetism. 
Second Series : — On the Number of Points at which a magnetic needle can take a 
position vertical to the Earth's surface. By Thomas Stephens Davies, Esq., F.R.S. 
L. 8c E. F.R.A.S. Royal Military Academy, Woolwich. 
Received January 28, — Read February 4, 1836. 
The plan and objects of this series of papers have been so far explained already as 
to render it superfluous to enumerate them here. 
At the close of my former paper* I have given the rectangular equation of the curve 
of verticity, — or that in any point of which a magnetic needle being placed, its line of 
direction would pass through the centre of the earth, and consequently be vertical to 
the horizon at the point where it cut the surface : but as the form and character of 
the curve could not be directly obtained from that equation, nor from any other into 
which it could be transformed ; and as, moreover, the process by which they could 
be obtained required considerable preliminary investigations, I preferred to leave it 
in that state rather than give the partial and incomplete solution, which, in the midst 
of the deep domestic affliction that I was involved in, I must then have done. It is here, 
however, by pursuing a different course fully determined, as far, at least, as it is subser- 
vient to our physical problem: and I did not feel myself at liberty to insert in a paper 
on that subject any collateral inquiries, which, however interesting in a geometrical 
point of view, would be irrelevant to the immediate discussion professedly before me. 
By transforming the rectangular equation (76.) of the curve of verticity into a 
polar one, I have shown that there are two values of the radius-vector, and only two, 
for every value of the polar angle ; and a few of the consequences which seemed likely 
to facilitate our inquiry are deduced from it. The genesis of the curve, however, 
pointed out the necessity of a more complete examination of the magnetic curve 
itself: and it will appear that even in a geometrical view, and independently of any 
of its physical applications, this latter curve (the magnetical) is amongst the most 
elegant and interesting we possess. The method of investigation is, as far as I know, 
a new one : but it is one that in many cases, besides the present, may be very effective, 
and therefore valuable. Still as I could not lay down the principles of the method in 
this paper, so as to justify my processes, I have so modified it by a combination with 
the method of rectangular coordinates as fully to answer my present purposes. The 
method consists in taking as the variables in the equation of the curve, the angles 
* Philosophical Transactions, 1835, p. 246, equation 76. 
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