RoGKUs — Mode of Representing Linear Orthogonal Transformation. 63 



6. Plane-normal flats. — This conception is important in what follows. If 

 the normal flat to (S',„ is a plane S.2 lying in Sm, and the normal flat to *S^„,' is a 

 plane S\ lying in S,,,, then S,,, and Sm are said to be inutxudly plane-normal. 

 Evidently m = m' = n - 2. Each of the flats is termed plane-normal. 

 Thus the two flats (x =0, y = 0) and (v = 0, w = 0) are mntually plane- 

 normal, and if n = 5 (for example) the corresponding normal planes are 

 (s = i(, = V = w = 0) and (x = y = s = u = 0). 



A plane-normal flat S,i-i has 2 (n - 1) co-ordinates. 



7. A complete system of mutually plane-normal flats is one containing 

 the greatest possible nimiber. Now let ^S be plane-normal to a given ^S, say 

 a; = 0, y = 0. Then the coefficients of x and y in the two equations for ^S 

 must vanish, and this reduces the number of its free co-ordinates by 4. And 

 if it is plane-normal to A'' given flats the number of its free co-ordinates is 

 reduced to 2{n - 1) - iN. 



Hence, if n is even, a complete system contains — mutually plane-normal 



n - 1 

 flats ; if nis odd, the niLmher is ■ — ^ . Thus if n = 6, the flats {x = 0, 



y = 0), [z = 0, 16 = 0), {v = 0, w = 0) form a complete system ; if n = 5, 

 then (x = 0, y = 0), {z = 0, m = 0) form a complete system. 



8. This leads to a principle which is the keystone of the argument : — A 



complete system of mutually plane-norm,al flats has -^ or — ^ — free co-ordinates, 



according as n is even or odd.* Let ^Sm, ~Sm, ^Sm, etc., be such a system. '*S',„ has 

 2(n - 1) co-ordinates ; when ^S,n is fixed, 'Sm has only 2(n - 1) - 4 free co-ordi- 

 nates; when both are fixed, ^S,„ has 2{n - 1) - 8 free co-ordinates. Thus the 

 total number of co-ordinates of the system is the sum of the series 



2{n - 1) + 2(w - 1) - 4 + 2{n - 1) - 8 + 2(w - 1) - 12 + 



7^ or — ^ — terms, according as ?i is even or odd, and is therefore 



w- n' - 1 



2 °' -^— 



When H = 3 this is replaced by the principle that a right line has four co-ordinate.s 



