swinging through an Arc of Finite Magnitude. 859 



7r — am(u + v). We may call the triangle so formed 

 Legend re's triangle, because it is really the triangle in- 

 vestigated by Legendre, though obtained in a manner 

 different from his. By applying the ordinary formulas 

 of Spherical Trigonometry in Legendre's fashion to this 

 triangle we easily reach the addition-theorem ; the algebraic 

 details are well known, and need not be repeated here. 



The two Eccentric Circles of Jacobi. 



We shall now bring the above investigation into con- 

 nexion with the geometry invented by Jacobi to illustrate 

 the addition-theorem. 



Let AB be the minor axis of a sphero-conic of the nature 

 already described ; take the tangent-plane to the sphere at 

 the point A, and project the spherical figure from the centre 

 of the sphere onto this tangent plane. The point A is 

 evidently unaltered by the projection ; the projection of B 

 may be denoted by B', and the arc AB will then project 



into the straight line AB 7 . The sphero-conic will project 

 into a circle on AB' as diameter, since the tangent-plane 

 at A is parallel to one of the cyclic arcs of the sphero-conic. 

 The other cyclic arc of the sphero-conic will project into a 

 straight line lying outside the circle ; let us denote this 

 straight line by HK. By symmetry, HK will cut B'A 

 produced at right angles ; and it' (x denote the point of 

 intersection, we shall have G closer to A than to B'. The 

 straight line B'G subtends a right angle at the centre of the 

 sphere. 



3 K 2 



