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



through the points where the conic ^ = 0, a;^ + ^^ + 2^ = is met 

 by the hne ax+hy + cz=0. The transformation 



{x', y', z', t') = Q {X, y, z, t) 

 makes precisely the same changes in the parameters q, f respec- 

 tively. 



Clearly the representative point {a, b, c, d) is the point to which 

 the vertex (0, 0, 0, 1) is changed by the transformation P. This 

 definition is relative to the position chosen for this vertex upon 

 the axis of the original transformation, but this is immaterial^ in 

 discussing the composition of rotations about different axes passing 

 through the same vertex. 



In particular the representative points for rotations of ampli- 

 tude 77 respectively about the lines joining (0001) to (1000), to 

 (0100) and to (0010), are (1000), (0100), (0010), the corresponding 

 transformations P being 



0,0, 0,1 , j^ 0, 0,1,0 , k== I 0,-1, 0,0 



0,0,-1,0 0, 0,0,1 I 1, 0, 0,0 



0,1, 0,0 -1, 0,0,0 lo, 0, 0,1 



-1,0, 0,0 0, -1,0,0 |0, 0,-1,0 



by which the point (x, y, z, t) becomes changed respectively to 



{x'= t, y'= — z, z'^ y, t'^ — x), (a;'= z, y'^ t, z'= — x,t' = — y), 



{x'= — y, y'^ X, 2'= t, t'= — z). 

 We at once find 

 ^2 = J2 = y^2 = _ ^^ jj^ = — kj =^ i, hi = — ik — j, ij = — ji = k, 



and the matrix of the general transformation P can be written in 

 terms of the matrices i, j, k in the form 



P = ai + bj + ck -{- do), 



where co is the matrix of the identical transformation. For the 

 generators of the ^^-system, the transformations i, j, k lead re- 

 spectively to p'= p~^, p'= — p~^, p'= — p, that is they are har- 

 monic inversions in the pairs of generators of the p-system passing 

 through the intersections of the conic x^ + y^ -\- z^ = 0, t = re- 

 spectively with the lines a; = 0, ^ = 0, 2; = 0. By the transforma- 

 tion P, these three pairs of generators are changed to the generators 

 passing through the intersection of the conic respectively with the 

 lines 



ux + h^y + gz = 0, hx + vy +fyZ = and g-^x + fy -f wz = 0, 



as is easy to see. For instance the generators p = 1, p = — 1, are 

 changed respectively to the generators 



p = {u + ihj)l{l +g), p = -{u + ihj)/{l - g). 



