185 



v are the same in the equations of the two surfaces ; but at the 



origin - = 0, - = 0, i =0. At this point let =0, v=v//. Sub- 

 v 4 



stituting these values in the preceding equations and dividing by = oo, 



and as these equations represent the same tangent plane, they must 

 be identical. Hence we shall have, introducing an equalizing factor A, 



a=Xa, a,=Xa /} a n =\a lt , 6=X/3, ,=X/3,, b tt \ft n . 

 Making these substitutions in the preceding equations, they become 



Multiplying the former equation by X, and subtracting from it the 

 latter, we get 



tbe tangential equation of a point which is the vertex of the second 

 enveloping cone. 



Now the protective coordinates or the xyz of this point are 



Again : as at the beginning of this abstract we assumed the well- 

 known property that three confocal surfaces of the second order 

 which meet in a point intersect each other at right angles, so if a 

 tangent plane be drawn to three concyclic surfaces, the three points 

 of contact, two by two, will subtend right angles at the centre. The 

 proof of this is very simple. Let the tangential equations of two 

 concyclic surfaces be 



Subtract these equations one from the other, and we shall have 



And as these surfaces are concyclic, 



