Royal Society. 419 
In his former paper the author had shown that on this hypothesis 
the magnetic equator, or the locus of the points at which the mag- 
netic needJe takes a horizontal position, is one single and continuous 
line on the surface of the earth. In this paper his object is to prove 
that there are always two, and never more than two, points at the 
earth’s surface, at which the needle takes a position vertical to the 
horizon. 
At the close of his former paper the author had deduced the 
equation of the curve of verticity, that is, of the curve at any point 
of which an infinitesimal needle being placed, it will always tend 
towards the centre of the earth, and consequently be vertical to the 
horizon at its point of intersection with the surface of the earth - 
but, owing to circumstances over which he had no control, he was 
unable, at that time, to write out an account of his investigations of 
the peculiar character of that curve, or to apply its properties to the 
determination of the latter problem: and these are more especially 
the objects to which the present paper is devoted. 
The processes to which he has had recourse, with this view, are 
the following. He first transforms the rectangular equation of the 
curve into a polar equation, and finds that in the result the radius 
vector is involved only in the second degree; and hence that for 
every value of the polar angle there are two values of the radius 
vector, and never more than two ; or, in other words, that no line 
drawn from the centre of the earth can cut the curve of verticity in 
more than two points. But as no means present themselves of ase 
certaining whether the values of (r), the polar ordinates of the curve 
of contact, be always real or not, or how many values of (6), the 
other co-ordinate to that curve, are possible for any given value of 
7; he abandons this method of inquiry, contenting himself with a 
few deductions respecting the general form of the locus, and proceeds 
to employ a different method. 
The general system of his reasonings proceeds on the principle 
that as the magnetic curve itself, and the curve of verticity have one 
common and dependent genesis, a knowledge of the properties of 
the former must throw considerable light on those of the latter; and 
he is accordingly induced to enter into a more minute examination 
of the magnetic curve than had before been attempted. As both 
the polar and the rectangular equations of this curve are much too 
complex to afford any hope of success in their investigation, the 
author has recourse to a system of co-ordinates, which he terms the 
‘angular system,” and which was suggested to him originally by the 
form under which Professor Playfair exhibited this equation in Ro. 
bison’s Mechanical Philosophy. But as he has not yet published 
his investigations of the differential coefficients, and other formule 
necessary in the application of this system, he puts his results in 
a form adapted to rectangular co-ordinates ; each rectangular co- 
ordinate being expressed in terms of his angular co-ordinates and 
the constants of the given equation; and by these means deduces 
the characters of the magnetic curve throughout its whole course. 
The angular equation being 
cos 6, + cos 6,, =2cos B, 
2U2 
