Prof. Baker, On transformations with an absolute quadric 145 



On transformations ivith an absolute quadric. By Professor 

 H. F. Baker. 



[Read 9 February 1920.] 



We consider homographic (linear) transformations of projective 

 space which leave unaltered a given quadric, sometimes called the 

 Absolute. We suppose the two generators, of either system, of 

 the quadric, which are unchanged by the transformation, to be 

 distinct. Denoting then by DC and AB the two diagonals of the 

 skew quadrilateral formed by these four generators, it is known 

 that the transformation may be represented {a) by rotations about 

 DC and AB of suitable amplitudes, whose order is indifferent, 

 (6) by a "half turn" about an arbitrary line meeting DC and AB, 

 followed by a half turn about another appropriately chosen line 

 meeting DC and AB, or preceded by such a half turn of appro- 

 priate axis, (c) by a "right vector" and a "left vector" together, 

 whose order is indifferent. The object of the present note is to 

 mention another mode of decomposition of the transformation. 

 For this purpose we define inversion, in regard to a point and a 

 plane w, as the process of passing from a point P to a point P' on the 

 line OP such that P, P' are harmonically separated by and w. 

 We consider only cases in which m is the polar plane of in regard 

 to the absolute quadric. We also define harmonic inversion in 

 regard to two given skew lines, as the process of passing from any 

 point P to a point P' on the transversal drawn from P to the lines, 

 such that P, P' are harmonically separated by these. When the 

 lines are polars of one another in regard to the absolute quadric, 

 the process is the same as a half turn about either of them, and is 

 obtainable as the sequence of two inversions about any two points 

 taken on either of the lines so as to be conjugate to one another 

 in regard to the quadric. 



Thus if H, K be two arbitrary points respectively on AB, DC, 

 and H^, K^ be two appropriately chosen points on these lines 

 respectively, it follows from the decomposition (6) referred to 

 above that the complete transformation can be represented by 

 the sequence of four inversions 



wherein, since H, K^ are conjugate points, the process represented 

 by {K-^ (H) is the same as that represented by {H) (Kj). The 

 whole is then equivalent to 



(H,) (H) . (A\) (K). 



VOL. XX. PART I. 10 



