290 

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



Hence, O T+ OT + (TF+ TQ) will be constant, iidfi^O, 

 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, ju, 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 (a) 

 may be regarded as the differential of themixtilineal 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.*</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 



