346 WILSON AND MOORE. 



8 = (eki + ^Ik3)cos20 + B'k.s^'m29, we see that B = represents the 

 general case. If Be 9^ 0, the indicatrLx (Conic I) is a true eUipse with 

 central radius 8. Referred to its center the equation of the indica- 

 trix is 



eV - 2AeziZz + {A'' + 52)^^2 = 52^,2^ 



as may be seen from (113). To find the locus of the intersection of 

 consecutive normal planes we need the inverse of the pedal of the 

 ellipse with respect to the origin. One observation may be made in 

 advance: the conic (Conic II) which will be found must contain the 

 origin in its interior. 



The calculation of the inverse pedal may be carried through neatly 

 by vectors. If be the selfconjugate two dimensional dyadic that 

 gives the conic referred to its center as 8«f2«8 = 1,^*8 is the normal, 

 and the equation of the tangent at 8, or at 8 -)- h when referred to the 

 origin, is 



(r - h - 8).r}.8 = or r-<7-8 = l + h-fi-8, 

 where r is the radius vector. Hence 



r-P-'8 ^ ^ Q'h 



^ = 1, and p = i. 



l + h-12-8 ^ l + h-12-8 



is the radius vector of the inverse of the pedal. Then 



1 - h-p (1 - h-p)2 



The inverse of the pedal is therefore, 



p.S2-i.p = 1 _ 2h-p + (h-p)-. 



Taking U from (114) with / = we find 9,~'^ at once and hence the 

 desired locus (Conic II) is ■ 



(e2 - h^)z,^ + 2Aez^^ + (.l^ + 52)^3^ + 2hz^ = 1. (116) 



The discriminant of the first three terms is ^^(^42 _|_ ^2^ _ j^2^2^ -pj^^ 

 conic is an ellipse, parabola, or l\yperbola according as this expression 



