Geometric Properties of the Magnetic Curve. 313 



may be a fourth proportional to the difference of the cubes of 

 C N and C S, the cube of C S, and the line N S *. 



The most remarkable property of the magnetic curve is, 

 that the difference of the cosines of the angles, which lines 

 drawn from any point in the curve to the two poles make with 

 the axis, taken on the same side, is a constant quantity. 



In order to demonstrate this proposition, we must take 

 another point, C', in the curve, Fig. 3. exceedingly near to the 



Fig. 3. 



former point C ; and from N and S draw to it the lines N C', 

 S C', which should be produced a little beyond C' ; and de- 

 scribe from the centres N and S, respectively, the small arcs 

 C A and C B to meet these lines. Draw also C E perpendi- 

 cular to N T. Let the same notation as before be preserved 

 with regard to w, s, p and q. 



* The following is a convenient way of obtaining, geometrically, lines which 

 are in the ratio of the cubes of two given lines, A B and A C, Fig. 2. Set them, 



Fig. 2. 



as in the Figure, at right angles to one another. Join B C, and draw A D per. 

 pendicular to it. Draw C E, and D F, parallel to A B j and D E parallel to A C. 

 The lines D F, DK, will be to one another in the ratio of the cubes of AB 

 and AC. 



