189 

 may be reduced by a transformation of coordinates to the form 



The new origin being fixed at any point in space 

 (iTo Vo 2o)j the normals to the three confocal surfaces of the 

 second order passing through that point are made the new 

 axes of the rectangular coordinates, ^, tj, Z,. The quantities 

 k^, P, /i"^ are determined by the equations ¥ = a? - a^, 

 f^i = a'3 _ a^2^ k"-2 = a"2 - tto^, in which a^, a'^, a"-\ are the squares 

 of the primary semi-axes of the three confocal surfaces ; /stands 

 for the quantity 



■*o !/o ■^o 



Cto t)o Co 



and ^0, rjo, So, are the coordinates of the centre of the surface. 

 It has also been observed by the same geometer, that 



is the equation of the cone whose vertex is at the point 



{xo Vo Zo)> and which envelopes the surface (oo h Co) ; whilst 



lo ? ThJl tJ-. _ , 



is the equation of the plane of contact of the cone and surface. 

 From this form of the equation of the surface of the second 

 order we are enabled to deduce a general theorem ; the con- 

 sideration of a particular case of which suggests a simple proof 

 of Joachimsthal's theorem concerninggeodeticlines, pd= Const. 

 If a perpendicular p he let fall from the centre of a sur- 

 face of the second order (oSo h Co) ttpon any tangent plane to a 

 cone enveloping the surface, we shall have 



I _ Cos^g Cos^^ Cos^Y , 



WJ} " {.a? - ao^f ^ {a" - Uo^f ^ (o"^ - ajf ' 

 where L is the length of the side of contact ; a, (i, y, are 

 the angles it makes with the axes of the cone ; and a, a', «" are 



