PROFESSOE C. J. JOEY ON QUATERNIONS AND PROJECTIVE GEOMETRY: irlSS 
88 . When dg', instead of being perfectly arbitrary, satisfies 
dQ = 0 or Sy> dg = 0, where Q = 0 .(363), 
dq rej^resents some jDoint in the tangent plane at q to the surface Q = 0. The j)oint 
p is the reciprocal of the tangent plane with respect to the unit sphere =’ 0; and 
the surface P = 0 is the reciprocal of the given surface. The relations of reciprocity 
are clearly exhibited by the equations (compare (354)) 
Sy» dq = 0, Sq dp = 0, dP = 0 if Q = 0, clQ = 0 . . . (364) ; 
— S dqjdq = Sp d’-q = Sq d^p, d^P 0 if also cPQ = 0 . . (365). 
89. For the asymptotic lines, in addition to (364) and (365), the new relation 
0 = S dy> d(^ = Sp d^q = Sq d^qy.(366) ; 
and thus for arbitrary scalars x and y 
S (p + ^dp) (q + P dq) = 0 .(367), 
or the reciprocal of an asymj^totic tangent is the asymptotic tangent to the reciprocal 
surface at the corresponding point. Hence also, if corresponding tangents are 
reciprocal they touch asymptotic lines. 
The tangents to the asymptotic lines of the original surface are also represented .by 
the equations 
Srjr = 0, Spr = 0 .(368) ; 
and those of the reciprocal surface by 
S?'qr = 0, Sq?- = 0.(369); 
r being allowed to vary arbitrarily, but p and q being kept constant. These lines 
are, in fact, the generators of the reciprocal quadrics 
Srfr = 0 or Srg~h’ = 0, and Srgi‘ = 0 or Srf~h' = 0 . (370), 
(compare (362)) which lie in the corresponding tangent planes. 
90. The generalized normal to a surface at any point is the line joining that point 
to the pole of the tangent plane with respect to the quadric of reciprocation. But 
as there is j^ractically no additional labour involved in the following discussion when 
the auxiliary quadric is arbitrarily selected, we assume it to be 
Sqhq = 0 . ..(371) ; 
and then the equation of the normal at q to the surface Q = 0 is 
r = q tli~^p ..(372). 
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