THE DYADICS IN THREE DIMENSIONS. 417 



The points that are transformed into lines therefore lie on the quadric Q, 

 whose equation is 



^{qiq,){fi,X)ifikX) = 0. 



Points on a generator of the quadric transform into lines any one 

 of which is a linear function of any two others. Such a system of 

 lines is a flat pencil. Hence the points of a generator transform into 

 the lines of a flat pencil. If the quadric is not singular, take a skew 

 quadrilateral on it and let /3i, ^-2, (83, (84 be the planes determined by 

 consecutive sides. Then q\, q-i, 73, qi will be lines for they correspond 

 to the vertices of the quadrilateral (which are points of Q). Further- 

 more, qi and qi belong to a pencil and so intersect. Similarly qi 

 intersects qs etc. The four lines therefore form a quadrilateral. 

 If / is a diagonal of the quadrilateral it cuts all the lines qi. Hence 



im = 0. 



The diagonals are therefore the same (being the singular lines of the 

 dyadic) whatever quadrilateral ^i, ^2, ^3, jSi is taken on the quadric Q. 

 The flat pencils corresponding to points on a generator have their 

 vertices on one of those diagonals and their planes pass through the 

 other. All the generators of one system of Q give pencils with vertices 

 on one diagonal, all those of the other system give pencils with vertices 

 on the other diagonal. 



Points of a plane transform into complexes with one system of 

 generators on a quadric R in common. Points on the intersection of 

 the plane with Q transform into the other system of generators of R. 

 If the plane is tangent to Q, points on the intersection transform into 

 lines of two plane pencils. 



If Q degenerates into a cone, points on a generator still transform 

 into the lines of a pencil. In case of the general quadric, the pencils 

 corresponding to one system of generators have vertices on one line, 

 those corresponding to the other system have vertices on another. 

 In case of the cone, the two systems of generators coincide. Hence, 

 the two lines on which the vertices lie, coincide. This line belongs to 

 all the pencils and so is the transform of the vertex of the cone. In 

 this case the four complexes qi, q^, qz, qi have only one line in common. 



Two points X, Y are harmonic with respect to Q if 



i:{qiqkWiX){§uY) + {fiiY){^uX)\ = 0. 

 Since 



i:qSi= ^qk^k = $ = g/3 



