of Curves^ 'wf4^0i?pgd'h'it>-J^&/^ 6f ike^ \S^dkd^brder. si^ 



XX. We shall find that peculiar relatiort^JeSi§¥^\V^H" 

 conic sections having the same minor directl*l<!^,''#9rt)^llat 

 analogous to those of confocal conic sections; wd'^M^M^hfi'^ 

 the present occasion give more than a fevvof them. " 'L'y^^/t'b^" 

 the distance between the common centre and one of the dir^^ 

 trice%.then ithe axes are connected by the relaftc^^ °^ nwBib) 



A Ai.^ 1 wo rectangular radn vectores bemg drawn from the 

 coriimon centre of two conic sections having the same minor 

 directrices, one to each section; the sum of the squares of the 

 reciprocals of those semi-diameters is constant, and the line 

 which joins their points of intersection with the curves enve- 

 lopes a circle ; and if tangents are drawn to the curves through 

 the points where the central radii vectores intersect them, the 

 locas of the intersection of those tangents will be a conic sec*- ' 

 tion, having the same centre and minor directrices a&<jtiKfe*i 

 former. o£|t fbtdv/ mo-il JbHj oJ guonolcnfi si mo'iosdi airiT 



XXII. Let a common tangent be drawn to two'^eonic sec- ' 

 tions having the same minor directrices, the line connecting = 

 the two points of contact subtends a right angle at the centre. 



XXIII. From any point in one of the minor directrices let 

 a series of pairs of tangents be drawn to the conic sections, all 

 the cords of contact will meet in a point on this directrix ; or 

 more generally, let there be a series of sections having the 

 same centre and minor directrices, and from a point in the 

 external one, let paii's of tangents be drawn to each of the in- 

 ternal sections; the cords of contact will all meet in a point 

 on the tangent to the external sectio|i dmtvn trough the 

 given point. lonim sril douoJ o^Ii? iUw ^a ^1 ,9*1 



XXIV. The'difl^Miifie'of tK^*%^Ve§'i6nR^-V^Hprocals of 

 any two coincident semi-diameters of two conic sections having • 

 the same minor directrices is constant. 



XXV. Let a series of conic sections, all having the same 

 minor directrices, be cut by a transversal ; the segments of 

 this line between any pair of sections subtend equal angles at 

 the centre, and if through every pair of points in which tliisf'*^ 

 line intersects the sections, tangents are drawn intercepted^ '' 

 both ways by the directrices, the sum of the angles which any 



