PROFESSOR C. J. JOEY ON QUATERNIONS AND PROJECTIVE GEOMETRY. 281 
= 0 , ^qf,q = 0 , Bqf^q = 0 , ( 346 ) ; 
take the polar planes of a point q with respect to the first and second, and the poles 
of these planes Math respect to the reciprocal of the third; these poles are conjugate 
to that reciprocal provided the point lies iqDon the quadric 
= 0.(347). 
If the quadrics have a common self-conjugate tetrahedron with the cjuadric of 
recq3rocation, the three functions have the same united points, and are consequently 
commutative; and the three surfaces (347) obtainable for different selections of the 
quadrics (346) are identical. 
84. Before leaving this subject, it may be of interest to show how the invariant 
condition that three quadrics should be polar quadrics of a cubic presents itself 
We have, if the quadrics are polars of the cubic F(qqq) = 0, 
= F {aqq), ^qj\q = F {hqq), ^qf^q = F {cqq) 
if a, h, c are the poles. Hence 
SqfJj = Sqfoa ; Sqf.C = Sqf^b ; Sqf/t = Sqf^C 
and on identifying the planes 
J\b =f.a; f.c =f^h; J\a=f^c . . . 
80 that 
«=y'rV'i/3‘'A/r'/3«. 
( 348 ), 
(349); 
( 350 ) ; 
( 351 ) ; 
and the function V1/3 V2/1 have one latent root equal to unity. 
SECTION XIV. 
Peoperties of the General Surface. 
Art. Page 
85. The principle of reciprocity, Q = Sj ;(7 = P.281 
86 . The self-conjugate function / defined by cG = (?« - 1) / (dq) .282 
87. The reciprocal relation= 1, \yhere 012= (ft - 1) (dji).282 
88 . The relations of recfiirocity, - Sdyd 2 = Sj)d “2 = S 2 d 2 /i.283 
89. The reciprocal of an asymptotic tangent is an asymptotic tangent to the reciprocal. . 283 
90. Generalized normals and centres of curvatui’e.283 
91. The osculating quadric and its confocals.284 
92. The quadratic equation of the principal curvatures.285 
93. Generalized geodesics.285 
85. If Q is a homogeneous and scalar function of a variable f|uaternion q of order 
m, the equation 
Q = 0 
(352) 
VOL. CCI.—A. 
2 o 
