546 Dr. Booth on a Neio Class of Properties of Curves, 



pair of these tangents drawn to the same conic subtends at 

 the centre is constant. 



XXyi. Let a right line Q Q', and a point O be assumed, 

 and round the latter as pole let a constant angle revolve whose 

 sides meet the given line in the variable points Q, Q'; let two 

 other fixed points P P' be assumed, the locus of the intersec- 

 tion of the lines P Q, P'Q' is a conic section, which passes 

 through the points P P', and of which (when O is the centre) 

 the given right line is a minor directrix, 



XXVII. Let a series of concentric conic sections, having 

 the same minor directrices, be cut by a common diameter, the 

 tangents drawn through the points where this diameter inter- 

 sects the curves, envelope a concentric conic section. 



The solution of this question is so simply obtained by the 

 method of tangential co-ordinates, that I am induced to give 

 it here. Let ...^c,, :>j cj.-..^.ii ^.s.. ......i.- ..•. 



iBlimie fnog2y2 -|_ ^2^ =dip^ IipiijDdla o,w}.6T[^ .^) 



be the tangential equation of one of the series of ellipses or 

 hyperbolas, and as they all have the same minor directrices, 



^.., „. .... ±', ±3^ Ji, , ^ . ^ ._-.....- -^^. 



-9ra Jnsbndqaboi oa^ h^ ^^ij'irij^uoa ayitfl'l ^oifiw raafld 

 let^*4a4i^1be the projective equation of tlie diameter j then 



cJasnua sdi jiaiiivv <sic — ^- — ^ (3.) 



. sdti*.i"F'il :3'b 01 as ni yrp ^ ^ 



Eliminating a and b between the equations (1.), (2.), and (5^)^"^^ 



we find - • • - .. -aaffi 



tneequanon'^oi a' concentric equilateral hyperbola or ellipse. 



In an early number we hope to return to this subject, and 

 apply this theory not only to oblate spheroids and central sur- 

 faces of revolution generally, but to surfaces of the second 

 order having three unequal axes, as also to systems of sur- 

 faces having coincident circular sections, groups which bear a 

 striking analogy in their relations to systems of conic sections 

 haying the same minor directrices; showing among other re- 

 markable properties, that every surface of the second order 

 hasfour directrix planes parallel two by two respectively to 

 the circular sections of the surface, as also four foci situated 

 two by two on the umbilical diameters. 



\A 83on9b8 sab seiobusY hh'v: 



- '""«f;NL.ii?»irfqoaoJifW[ aril ai ^, _ .., — ...... 



.h^ .q .vi .ioyf ^i^\^^iKis^\ «b t^tmVjtK arid! ni bas Jiy&^ .^ «t4^ Ut>v 



