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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 transposes 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 as- 

 certaining whether the values of (r), the polar ordinates of the curve 

 of contact, be always real or not, or how many values of (0 y ), the 

 other co-ordinate to that curve, are possible for any given value of 

 r; 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 formulae 

 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 ; X cos S H = 2 cos /3, 

 he finds, 1°, that the two equations, the convergent and the divergent, 

 or that in which the poles are unlike, and that in which they are 

 like, are both expressed by this equation, and essentially included 

 in it : c 2°, that the divergent branches on one side of the magnetic 

 axis are algebraically and geometrically continuous with the con- 

 vergent branches on the other side; the parameter (/3) being the 

 same in both cases : 3°, that the divergent branches are assym- 

 ptotic, and the assymptote is capable of a very simple construction ; 



