Rogers — Mode of Representing Linear Orthogonal Transjormation. 65 



A rotatory flat is one round which rotation is possible. 



A rotatory flat must he an 8,1.1- For Sm has {11 - m) {m + 1) co-ordinates 



which fix it geometrically, and there are „ degrees of freedom of 



motion within S,,,, Hence if S,n is kept rigid, the number of restrictions is 

 the sum of these numbers, viz. : 



{2n - m) (in + 1) 

 2 



But by definition this is one less than the number of degrees of freedom in 



_ . m{m+V) 



ISm, VIZ. — - ■ Hence m = n - z. 



12. Co-ordinates of a rotation. — A rotatory flat /S„_-j has 2{n - 1) co- 

 ordinates. Hence a rotation has 2n - 1 co-ordinates, or independent parameters 

 ivhich specify it. 



13. In a rotation round (S,,... the other points in S„ must move in circles 

 whose centres lie in 8^2 and whose planes are normal to Sn.2. For, let S„.i 

 be (x = y = 0). When x = 0,y = 0, there is no change in any of the co-ordinates, 

 hence x' = l-^x + m^y, y = l^ + miy ; 



and applying the conditions of orthogonality the transformation must be 

 expressible in the form 



x' = X cos 8 + y sin 0, 



y' = - X sin 6 + y cos d, 



z' = z, 



u' = u, 



etc. 



Thus any point (P), x, y, z, 10, v . . . . describes a circle round the foot (M), 

 0, 0, y, z, u, V, of the single perpendicular line through F to /S„_2- Also if P 

 moves to P' the angle PMP' = 0, and is the same for all points. 



14. A rotation may equally well be described as being mi an (n - 2) ply 

 infinite series of parallel planes, the planes normal to /S,j_2- These planes are 

 fixed, but not rigid. One point is fixed in each plane, the centre of rotation 

 for the motion in that plane, and <S„.2 is the locus of these points. Thus in 

 the rotation of § 13, the parallel planes are all of the form z = c, v = d, w =■■ e, 

 etc., where c, d, e are constants. 



15. General formula for rotation round Sn-2- — Let 6'„.2 be a fiat through 

 the origin containing the n - 2 mutually orthogonal directions 



{Ir, rii,; n,; pr ■ ■ ■ •)• 



