Prof. Baker, On a set of transformations of rectangular axes 151 



joining A, B to Q,n arbitrary point D of DC both unchanged, and 

 also the lines joining A, B to a, further arbitrary point C of this 

 line; referred to A, B, C, D, such a transformation will be ex- 

 pressible by equations x'= mX~^x, y'= mXy, z'= z, t'= t. If it be 

 further restricted to be such as to leave any (and therefore every) 

 quadric xij = Mzt through AD, BD, AC, BC, unchanged, we may 

 take m = 1. Then any conic through A, B touching the planes 

 ADC, BDC is also unchanged, and the equations of transformation 

 x'= )r^x, y'= Xy, z'= z, t'=^ t are those just considered. By the 

 component transformation x^ ^ x, ?/i = Xy, z-^ = Xz, t^ = t, any 

 point of DC is changed to a point 0^ of DC given by 



(OjO, DC) - A. 



In more general terms, if we write, using matrices, 



P = d, — c, b, a 



c, d, —a, b 



— b, a, d, c 



— a, — b, — c, d 



, Q 



d, -c, 



c, d, 



-b, a, 



a, b, 



b, 

 a, 

 d, 

 c, 



— a 

 -b 



— c 

 d 



denoting by P', Q' what P, Q become by substituting a', b', c', d' 

 for a, b, c, d, we find PQ'= Q'P, in particular PQ = QP, and 



PQ^ 



u, 



K 



9, 

 0, 



h, 



fi, 



0, 



9i, 

 f, 

 w, 



0, 



It is then easy to verify that the transformation 



{x', y', z', t') = P {x, y, z, t) 

 changes the f, g^-generators oi x^ + y^ + z^ + t^ = expressed by 

 X + iy = 'p {z -\- it), X — iy =^ — f-^ {z — it) ; 

 X + iy = q (z — it), x — iy = — q~^ (z + it) 

 into p', q' given by 



- J) (c — id) + a + ib 



V 



?; 



f {a — ib) + c + id ' 



this, with a = I sin ^6, b = m sin ^9, c = n sin ^6, d = cos ^6, is 

 equivalent with 



, I + im I + im 



V - T^— ,; P 



e'' - 



1 + n 



1 + n 



f + 



I + im 



V + 



I + im' 



\ — n ^ ' \ — n 



the stationary values of f being those of the generators passing 



