RoGEKS — Mode of Representing Linear Orthogonal Transformation. 61 



Before entering ou preliminary explanations, I will at once state that 

 for n dimensions the corresponding general principles are : — 



If 11 is even, a rigid schema can ie moved from one given position to another 



Oh 7i/ 



iy ^ independent rotations ro2i7ul 7, nniquel'tf determinable mutually plane- 



n - 1 

 normal [n - 2)-fl.ats ; if n is odd, hy — — such rotations, together v:ith an 



independent uniform translation along the common line of intersection of the 

 (11 - 2)-flats. The " independence " means that the order of the processes 

 is indifferent. 



2. A Euclidean manifold of n dimensions is the same as a. flat or space of 

 zero curvature, and is called Euclidean because the axioms of congruence 

 and Euclid's axiom of parallels are obeyed in all its geodesic surfaces 

 (planes). The element of distance may be expressed in n co-ordinates 

 X, y, z, u, V, w, etc., in the form 



ds^ = :^dx^\ 



and hence by the calculus of variations the shortest distance or the distance 

 between two points P^, P, i^ given by 



P,P/ = S (si - x^)^. 



The manifold will be described as an "?i-flat" or S„, and it will be 

 assumed that all other fiats are within <S'„. 



An 7ft-flat 8,„ is defined by n - m independent linear equations among 

 the co-ordinates, and is determined by m + 1 points. 



Every 7?i-flat consists of a singly infinite series of {m - l)-flats. Thus 

 the straight line Si is the fundamental flat. 



3. Orientation of fiats.— A. straight liue is determined by two points 

 (Pi, Po) by ''^ equations 



x-x^ = \ {x^ - a-j), y - yi = X (2/1 - 2/2)' etc., 



where A is arbitrary. The "direction cosines" of a straight line are n 

 quantities I, m, n, p, q, etc., which satisfy 



SZ^ = 1, 



and are proportional to x.^ - x^, y^ ~ y^, etc. Thus a line is determined by a 

 point P, and its orientation {I, m, etc.) by 



^-^1 y -y-L 



I m 



= etc. 



* See, however, Note, p. 73. 



[7*] 



