( 312 ) 



0, thus a conic on (f, thus the system of the tangent planes .t 

 passing throngli a fixed ])oint T. Through T and a |)oint X of tlie 

 base-curve of {(2'^) two i)lanes nr pass; wherefore A' hears two conies 

 of the congruence, which is thus of ordci- two {F = 2). 



3. To the tangent planes rt through an arbitrary point T corre- 

 s[)ond the j)oints /' of a conic not passing through J\, having thus 

 for image a conic in <P. So to this system (rr).^. of index two, is 

 conjugate a system (''/'').. possessing likewise index two, having two 

 surfaces in common with each pencil (Q^). When considering the 

 ranges of points determined by the projective systems (n:).^ and {Q-)^ 

 on an arbitrary right line we tin<l that they generate a surface 7'" 

 of degree six, which is the locus of the conies of the congruence 

 the i)lanes of Avhich pass throngli a lixed point 7'. Hence we get 



(I r :=: H. 



4. Through two arl)itrary points pass two tangent j»lanes rr, 

 hence the [tlanes of two conies; so an arbiti-ary right line is bisecant 

 of two conies, and the congruence is of c■^^v.v tiro {(r = 2). 



The numbers 7'*=2, jtr=zB and (i' = 2 satisfy the well known 

 formula P= (i v — 2 ƒ/*. 



Through a right line of Q- pas^ an inliniti' number of plajies .t; 

 the conies they beai' form a cubic surface. 



As each ray through 7' meets two conies, 7'" has in 7' a double 

 point. If .1' is one of the conies on which 7' is situated, 7'" is 

 touched in T by eacii bisecant of .1' out of 7'. So 7' is a biplanar 

 point. 



If T is ojie of the eight base points of the Jiet [(2''] then 

 7" has in T a fourfold point; for on every ray through T lie but 

 two points besides T. 



5. Let us take for Q' the ])araboloid .i- // = c, then the substitution 

 ,c = (( Q, // = (i (), -: = 7 (> furidshes lii-st ^ = y : « ,:? and then 



,v = V '• i^ ' !l = y ■• « 1 ~ = y' : i< i^- 

 So the tangent planes rr are represented by 



ji Y .1- -}- « 7 // -—<( t^ ^ — Y' -- 0. 



The above-indicated conjugation is arri\ed at l)y putting 



(( A + ,i K -f 7 C = U, 



where A, B, C are quadratic functions of ./;, //, :. We represent their 



coefficients by tt/,i, h/d , c/,i and we write brietly 



dl;i — a an -J- |3 hj: -f y cj. , 



