290 

 and that of the mixtilineal line O T'+ T Q will be 



Hence, O T+ OT + (TF+ TQ) will be constant, if dfi = 0, 

 that is to say, if O describe an ellipse confocal with the given 

 one. 



" We may now pass to the consideration of the analogous 

 theorem for the dimensions. The differential of a right line, 

 common tangent to two confocal surfaces of the second degree, 

 is in elliptic co-ordinates p, fi, v, 



b, c being the well-known constants in this system of co-or- 

 dinates, and a, j3, the parameters which determine the two sur- 

 faces above mentioned. Let one of them be defined by the 

 equation p=a, and the sum of the second and third terms, in 

 the foregoing expression, will be the differential of a geodesic 

 line traced upon this surface. Hence, the expression (o) 

 may be regarded as the differential of the mixtilineal line, com- 

 posed of a geodesic line ( G) upon the surface p = a, counted 

 from a fixed point, and of the linear tangent to it at its variable 

 extremity. And if we consider the geodesic line (G) upon 

 the same surface for which the coefficients of dp. and dv are 

 of opposite signs from those in the case of the line ( G), we 

 may easily see that if the sum of two arcs of the lines G and 

 G', counted from fixed points, together with the lengths of 

 the tangents applied to them at their variable extremities 

 (which obviously intersect), be constant, the locus of their in- 

 tersection will lie in a surface having for equation t/p = 0, that 

 is to say, a surface confocal with the given one. The same 

 expression (a) leads, with equal facility, to a theorem of M. 

 Chasles respecting lines of curvature. In order to obtain this 



