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



gra'pJiy; so that P, Q may be described as equal transformations 

 of the 23-kind and g-kind respectively (or if q' be interchanged 

 with q as equal but opposite transformations). 



If two lines p, q meet in A, two others p', q' meet in B, while 

 p, q' meet in D and j/, q meet in C, and on the transversal TLL', of 

 p, p', be taken T^ so that {T^T, LL') = A, and on the transversal 

 T^MM', of q, q', be taken T so that {T'T-^, MM') = A, it is at once 

 shown that, referred to ABCD, the coordinates of T', (x', y' , z', t'), 

 are expressible in terms of {x, y, z, t), the coordinates of T, by the 

 formulae 



x'= X, y'=^ X^y, z'= \z, t'=^ Xt. 



These are the equations of a rotation round DC, of which . every 

 point remains unaltered, as does every plane through AB. If the 

 planes joining DC to T' , T meet AB in tj' , U we have 



{U'U,AB) = XK 



The two processes may be taken in the reverse order. Any quadric 

 containing the lines p, q, p', q', whose equation is xy = Mzt, is 

 unaltered by the composite transformation. The transformation 

 here taken first is x-^ = x, y^^ Xy, % = Az, t^ = t, which changes 

 the p, g-generators, expressed respectively by 2; = px, y = Mpt and 

 y = qz, qx = Mt, into the generators pj^, q-^ given by p^ = Xp, q-^ = q. 

 The transformation here taken second is 



x'= ccj, y'^ Xy^, z'^ z^, t'= Xt-^, 



which changes p-^, q^ into p' , q' given by p'^ p-^, q'= Xq-^. 



Conversely any homographic transformation of space which 

 leaves every point of a line DC unchanged, and leaves also two 

 points A, B, not on DC, both unchanged, will leave the lines 



