Rogers — Mode of Representing Linear Orthogonal Transformation. 67 



When n is even, the number of flats in Fn is -^, and this is also the 

 number of angles in i?„. But the number of co-ordinates, of i^„ is -; hence 



LI 



the number of free co-ordinates of i?,, is 



'n? n n (71 + 1 ) 



— + _ = — ■ ■ 



2 2 2 



Now this is the number of degrees of freedom of motion in /S'„ (§ 9). 

 Hence : — 



When n is even, any displacement of a rigid schema can be effected by 



^ deternmiate rotations round ^ mutually plane-normcd {n - 2)-flats. 



n- -\ 

 When n is odd, the number of co-ordinates (§ 8) of F,, is — ^ — . But 



71-1 



there are — ^r~ flats in Rn and the same number of angles of rotation. 



Hence the number of free co-ordinates of an i?„ is 

 n^ - 1 71-1 n (n + 1) 

 2~ "^ ~2~~ " 2 ~ 

 Now this is one less than the number of degrees of freedom of motion in 

 Sn. Hence : — 



When n is odd, any displacement of a rigid schema can he effected by 



— ^ — determinate rotations roimd —^z — mutually plane-normal flats, com- 

 bined with a imiform translation along the common line of intersection of all of 

 these flats. 



17. Canonical form of the transformation. — The preceding amounts to 

 proving that the transformation of § 9 may, by referring the system to a new 

 set of mutually orthogonal axes, be reduced to the canonical form : — 

 X' = Zcos0+ Fsin6/, 



Y = -Xsiu(/+ Ycosti, 



Z' = ZcoB (f) + Usin <j), 



U' = - Zbiii (f> + i!7cos (^, 



V = Fcos^+ W sm\p, 



W = - F sin 1// + W cos <//, 



etc. 



When n is even, we have - such pairs of rotations. But when n is odd, the 



last equation, if T is the last co-ordinate, is 



T = T+t, 



where t is the uniform translation. 



