PEOFESSOK C. J. JOEY ON QUATERNIONS AND PEOJECTIVE OEOMETEY. 275 
The locus of the united lolanes of the system xf\ -\- yf 2 -\- z is the reciprocal of the 
conjugate sextic. 
By the conjugate sextic we mean the second curve (309), and the proposition is 
obvious when we reflect that a united plane of a function is the reciprocal of the 
corresponding united point of its conjugate (Art. 8). 
2 he united plane of a function of the system xj'^ ySz'f ^ caUs the sextic in three 
united points and in three other collinear points. 
The equation of a united plane of the function xf-^ + yf^^ + s is SAp = 0, where al is 
a united point of the conjugate. Writing the equation of the sextic in the form 
+ y'M -\-z'q = 0 . . . (310), 
and expressing that p lies in the plane, the result is 
S'i + y'ff^') — 0 , or Sp {{x' — sx) fja\ + jif — sy)ff/) — 0 
where s is arbitrary, because xf'a' + yfja' + za' ■= ta'. 
Hence either x' = x, f =: y, and p is a united point of the function, or else 
(311)> 
Sp«' = Sp// a' = Sp/oV =0.(312); 
and the three remaining points are collinear. 
In particular for the functions f fj /q, /, the polar plane with respect to the 
quadric and the plane of rays of the complex, corresj^onding to the reciprocal of a 
united jjlane of the function f, intersect in that united plane ; and their common line 
is a three-point chord of the sextic (306). 
74. Knowing the rank and number of apparent double points of the sextic, its 
characteristics are 
7- = 16, m =G,n= 30, a= 4:8, ^=0, x= 96, y = 72,g = 355, h=7 (313), 
as may be verified by the formula j^rinted in Arts. 326—7 of Salmon’s ‘ Geometry of 
Three Dimensions.’ Also the deficiency of the curve is D = 3. 
These numbers apply reciprocally to the developable of the last article generated 
by the united planes. Thus the order of its cuspidal curve is 30, and six united 
planes pass through an arbitrary point, while sixteen pass through a line. 
Thiough a united jiomt the six united planes consist of the three jilanes which are 
united planes of the function possessing the united point, and three other planes 
intersecting in a common line (compare (312)) which is the reciprocal of a three-point 
chord of the second sextic. 
75. The tripjle chords of the sextic generate a surface of the eighth order. 
The three-point chords of a curve generate a surface of order (‘ Three Dunensioiis,’ 
Art. 471) 
^ {m — 2) (6/i + m — nd) ........ (314), 
and this reduces to 8 in the present case. 
The characteristics of the cone, whose vertex is a point on the sextic and which 
contains the sextic, are deducible from the data of Art. 330 of the ‘ Geometry of Three 
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