350 SCIENCE PROGRESS 



Geometry. — In the general collineation of space the vertices, 

 planes, and edges of a tetrahedron are the only self-corre- 

 sponding elements ; but there are a number of special cases. 

 If all the points and lines of a plane a are self-corresponding, 

 so will be all the planes and rays through a point S, and the 

 collineation is a homology (or perspectivity). If all the points 

 of a line u and all the planes through a line v, skew to u, are 

 self-corresponding, the collineation is said to be axial ; but 

 if all the points on both u and v, and all the planes through 

 II and V are self-corresponding, the collineation is said to be 

 biaxial {gescharte Kollineation) . Moreover, the collineation 

 can only be involutory if it is a homology or biaxial. All this 

 is perfectly well known and is to be found in the textbooks, 

 such as Reye's Geometrie der Lage. But more recently (Reye, 

 Archiv d. Math., 19, 191 2, 288) intermediate cases have been 

 examined — non-involutory collineations containing involutions 

 of points, planes, or rays, which interchange the elements in 

 pairs. The square of such a collineation must be a homology 

 or axial or biaxial. For example, if a collineation interchange 

 two planes a-^, a^, it will interchange all the planes through their 

 line of intersection v in pairs ; the square of the collineation 

 will leave all planes through v invariant, and will therefore 

 leave all points of a line u skew to v invariant. The square 

 will thus be axial, and the original collineation may be called 

 half-axial. It will contain a point involution on u and a plane 

 involution through v. Similarly, we may have a half-biaxial 

 collineation {halb-gescharte Kollineation), which interchanges 

 in pairs the points and planes of two skew lines u and v. It 

 will also interchange in pairs the rays of the linear con- 

 gruence consisting of the double infinity of lines which meet 

 u and V. 



S. Jolles, who collaborated in the later editions of Reye's 

 book, has considered further cases of partial involutory 

 collineations {Math. Zs., 11, 1921, 180-93), and in a later paper 

 {ibid., 13, 1922, 223-62) discusses the theory of the pairing of 

 lines in a linear congruence by means of a half-biaxial collinea- 

 tion. Such a relationship is determined by two of the pairs 

 which do not belong to the same system of generators of a 

 quadric, and the pairs of lines are reciprocal polars for all the 

 quadrics of a pencil. There are four double rays, and we thus 

 get three species, according as the double rays are (i) all real ; 

 (2) two pairs of conjugate imaginary skew rays ; (3) two pairs 

 of conjugate imaginary lines of the first kind {i.e. each 

 with one real point). 



Prof. H. F. Baker has edited for the London Mathe- 

 matical Society {Proa. L.M.S., 21, 1922, 98-113) a remarkable 

 paper on " Chords of Twisted Cubics," by E. K. Wakeford, 



