RoGKES — Mode of Representing Linear Orthogonal Transformation. 73 



"We then I'otate the axes by the method of § 20, when the equations take the 

 canonical form, which is Ihe same for all origiiis on the central axis, the last 

 equation being T = t + T. 



We have further to prove that the central axis thus found is i.ormal to 

 all the {n - l)-flats ;■', y, z, u, . . . Tins amounts to proving 



SUr = 



for all values of S,. And it is proved in the same way as we proved 



n^t. = 0. 

 In fact (i. M, . . .) correspond to the unit root of f{X) = 1, and 



SliL = L, etc.; 

 also 2^1?,- = A,.?,., etc.; 



and therefore 2 L^r = 0, since A,- =1= 1. 



NOTE. 



A few days after this paper was read before the Academy, I had the good fortune 

 to come across Dr. Hilton's Homogeneous Linear Substitutions (Oxford, 1914), in which the 

 theory of orthogonal transformations is exhaustively treated. The theorem on p. 47 of 

 Dr. Hilton's work includes the algebra required to prove my kinematic conclusions in 

 the particular case when the origin is self-returning, and there is therefore no uniform 

 translation. 



I should add that I have not considered those rare singularities which arise (1) when 

 two or more of the angles of rotation are equal, and (2) when (n being even) one or more 



of the I rotations reduce to uniform translations (the rotatory flats being at infinity), 



n -\ 

 or (n being odd) one or more of the — ^-- rotations likewise reduce to uniform trans- 

 lations. Here Dr. Hilton's algebra will assist the inquirer who is fond of singularities. 



Those who are interested in the geometry of n dimensions will find ample references to 

 guide them in Dr. Sommerville's copious bibliography.* 



* Bibliography of Non-Euclidean Geometry, inchiding the Theory of Parallels, the 

 Foundations of Geometry, and Space of n, dimensions, by Duncan M. Y. Sommerville, M.A., 

 D.So. (St. Andrews University, 1911). 



K.I.A. PROC, VOL. XXXVI, SECT. A. [8] 



