CONCERNING TERRESTRIAL MAGNETISM. 239 



Again, since we can refer this surface, or any one of its meridional sections, to any 

 new system of coordinates, we may conceive the last equation (46.) to be so trans- 

 formed as to represent the same geometrical surface that it now does whilst it is re- 

 ferred to the centre of the earth as the origin, and to the mutual intersections of the 

 equator and two rectangular meridians as axes of coordinates. This transformation, 

 however, by the usual processes, would be extremely difficult — perhaps impracticable 

 — on account of the labour it would require ; but this labour may be almost wholly 

 avoided by means of the property (Playfair's) of the curve referred to the polar 

 angles expressed in equation (45.). 



Let the coordinates of the poles T and U referred to the above-named axes be 

 a^ b, Ci and a^^ b,, c^ respectively ; and the individual curve defined by the parameter ^. 

 Then viewing d^ and O^i as the two internal angles, we shall have 



^ {a;^a;)^ + (^- b;)' + {z-c;)' + (a,^a,;)^ + {b,-b,;)^ + {c,-c,;}'-(a;-a,;)'-{:y-b,^^^^ 

 2^ {(«,~«,)^ + (^',-6/ + (c,-c/} X {G^-«/ + (y-*/ + (^-c/} 



__ (a,-^) (a,-a,>) + (br-^) {b,-b,^ + {c,-z) {c,-c,^ 



and 



From (4.), (12.), and (13.), we have at once the equation of the surface, viz. 



(x - a,) {a, - g„) + (y - b,) (b, - b„) + {z - c,) (c, - c,;} 

 ^(a;^a,y+{t/-b,)^+{z-c,f 



_ (^ - a,) (a, - a,i) + ( j/ - &//) {b, - b,;) + (z - c,,) {c, - c,,) 



= ±2cos/3^/(a^-aJ2+(^_y2^(e«c,)2=:-f4«cosf3. 



This equation, deprived of its radicals and denominators, like the equation of the 

 generating curve, is of the eighth order. 



Now by varying the parameter j8 (which defines the particular surface upon which 

 the point we are considering is situated,) by minute increments, we shall have a series 

 of thin strata, each of which is isolated and independent, and which, collectively, ex- 

 tend through all space. If, therefore, we conceive these strata to be infinitesimally 

 thin, we may consider the magnetic influence extended over a series of surfaces, 

 eaeh of which has the property of being touched by a small needle placed at any point 

 in that surface. If, moreover, we conceive the sphere arid each of these surfaces to 

 be cut by a series of meridian planes passing through the magnetic axis, we shall 

 always find some one magnetic curve on each side of the magnetic axis which will 

 touch the two segments into which the magnetic axis (prolonged if necessary) divides 

 the circle lying in that magnetic meridian plane. At these points the needle will 

 touch the sphere, or, which is the same thing, it will be horizontal to the earth ; and 



> (49.) 



