314 Dr. Roget on the 



Let C E = e 

 CA = a 

 CB = b 



The angle C N T = 



The angle C S T = 



Cosine of a = c 



Cosine of /3 = x 



Then d cc = : and the triangle CAC' being similar to NPC 

 n 



n : p :: t : a; or a = t - 



n 



n : e :: da. : dc. 



j doc a , p 



dc = e = e e t J 

 n n* n* ' 



By a parity of reasoning dx = e t JL 



s 3 



Therefore dc:dx::JL:JL::R:r. 



w 3 s 3 



When R = r, d c = d x. Hence c = x + C, or c x = C. 

 That is, the difference between the cosines of the polar 

 angles C N T, C S T, is a constant quantity. 



When the angle C S T exceeds a right angle, its cosine 

 being then negative, the proposition will be changed to the 

 following ; namely, that the sum (instead of the difference) of 

 the cosines of the polar angles is constant. When the angle 

 C N T is also obtuse, both the cosines being negative, it is 

 again the difference of the cosines that is constant. 



The following method of describing this curve is derived 

 from the property above demonstrated. Let the two radii Nn, 

 S s, Fig. 4. be taken of equal length, and be made to revolve 

 in the same direction round their respective centres N and S, 

 while their other extremities n and s are always kept in such a 

 position relatively to each other, as that a Jne drawn through 

 them shall remain perpendicular to the axis N X ; then the 

 line constituted by the successive points of intersection C, c', 

 &c. of the two radii, will be a magnetic curve. This will 

 appear from the consideration that with the equal radii N?z 

 and S n, the cosines of the angles C N X and CSX are the 

 lines N X and S X, of which the difference is N S. In every 

 other position of the radii, as N ri, S s 1 , where the line s' n' x' 



