Prof. Baker, On a set of transformations of rectangular axes 147 



On a set of transformations of rectangular axes. By Professor 

 H. F. Baker. 



[Read 9 February 1920.] 



In a paper in the Acta Mathetnatica, xxv, 1902, 291-296, Dr 

 Burnside has greatly shnplified, by geometrical considerations, 

 results obtained in the same Journal, xxiv, 1901, 123-158, by 

 Lipschitz, for the relations connecting the four rotations changing 

 one system of orthogonal axes into another. In a paper. Proceedings 

 Loud. Math. Soc, ix, 1910, 197, I have incidentally noticed a 

 theorem which is intimately connected with these results, and 

 may be made to include them. We may associate with a rotation 

 in Euclidian space about an axis through the origin, of direction 

 cosines I, m, n, through an angle d, a point of projective space, of 

 coordinates {a, b, c, d) given by 



a = Z sin \d, b = m sin ^6, c = 7i sin 1$, d = cos 1$ ; 



and we may call this point, which determines the rotation, the 

 representative point of the rotation. The theorem referred to is then 

 the following: Let Q., Q.' be any two congruent figures upon a 

 sphere; let Q^, Q.^, O3 be the figures obtained from D. by reflexion 

 in, or rotations of amplitude tt about, the respective coordinate 

 axes; let Q/, Q^', ^3 be similarly derived from Q.'; there is then 

 a rotation changing any one of Q.^, O2, O3, O into any one of 

 ^1', ^2'' ^3'' ^'- ^^^ representative points of these sixteen rotations 

 form a Kummer configuration. In other words they lie in sixes 

 upon sixteen conies, whose planes are in sixes tangent planes of 

 sixteen quadric cones whose vertices are the points; or again, they 

 can be arranged in twenty ways so as to form the vertices of a 

 set of four tetrahedra of which set every two are mutually 

 inscribed. As will be seen, the theorem is the same as that, if a 

 point {a, b, c, d) be represented by the quaternion 



P = ai + bj + ch + d, 

 the sixteen points 



form a Kummer system, the four tetrahedra formed each by the 

 points in any row being mutually inscribed in pairs, as are the 

 four tetrahedra formed each by the points in any column. This 



10—2 



