Prof. Baker, On a set of transformations of rectangular axes 149 



But the definition of the representative point of a rotation 

 which has been given is, like this proof of the theorem, both 

 metrical and analytical, and it is desirable to alter it in both 

 respects. A point of view which is not metrical is to regard the 

 transformation 



x'=ux + hy + g^z, y' = h^x + vy + fz, z'= gx +fiy + wz, (I), 



as that unique homographic transformation of the plane of the 

 homogeneous variables x, y, z which leaves the conic 



x^ + y^ + z^ = 



unaltered, and changes the self-polar triangle (1, 0, 0), (0, 1, 0), 

 (0, 0, 1) into the self-polar triangle (n, h^, g), {h, v, fj), {gi, f, w). 

 If the vertices A, B, C oi the former be joined to the corresponding 

 vertices A', B', C of the latter, by lines forming a triangle D, E, F, 

 of which E, F will be collinear with A, A', etc., and we take on 

 the line A, A', E, F the points P, P' harmonic in regard both to 

 A, A' and E, F; and similarly take Q, Q' on the line B, B', F, D 

 harmonic in regard both to B, B' and F, D; and take R, R' on 

 the line C, C, D, E harmonic in regard both to C, C and D, E; 

 then it can be shown that the six points P, P', Q, Q', R, R' lie in 

 threes upon four straight lines, which are in fact 



dx + cy — bz = 0, dy -\- az — ex = 0, 



dz + bx — ay = 0, ax + by + cz = 0. 



This gives a geometrical interpretation, which is not metrical, of 

 a, b, c, d. 



A much better point of view is however as follows. The rotation, 

 expressed by four equations of which three are those marked (I) 

 above, and the fourth is t' = t, may be regarded as a homographic 

 transformation of projective space (x, y, z, t), leaving the quadric 

 whose equation is x^ + y^ + z^ + t^ = (or indeed any quadric 

 x^ + y^ + z^ + Mt^ = 0) unaltered. And it may be regarded as 

 compounded from two transformations P, Q, taken in either order, 

 of which P is such as to leave every generator of the quadric of 

 one system, say the ^--system, unaltered, while it interchanges the 

 generators of the other system, say the ^-system, among themselves ; 

 and Q has a similar meaning with the systems of generators inter- 

 changed. In the transformation P there will be two particular 

 generators of the /^-system, say p and p', which remain unaltered; 

 and the transformation may be described geometrically as changing 

 any point T of space, not necessarily on the quadric, into a point T' 

 of the transversal drawn from T to p, p', such that the homography 

 (T'T, pp') is constant, equal to e''^ say, the transformation Q 

 having a similar meaning in regard to two generators q, q' of the 

 g'-system and having the same value for the corresponding homo- 



