62 Proceedings of the Royal Irish Academy. 



The angle between two lines is 0, where 



cos d = 2//'.* 



Au S,n is likewise determined by a point P-^ and its orientation, which 

 may be taken as composed of the orientation (Jr, m,-, nr, etc.) of any in mutually 

 perpendicixlar lines lying in the fiat. The direction cosines (a, /8, y . . .) 

 of any line in Sm are given by n equations 



a = A/j + /i'rtlj + i'«i + . . . 



/3 = A/j + fl?»2 + l'«2 + • • • 



7, 8 ... = etc., 

 where 2A- = J. 



A flat with given orientation through P.^ is given by 



X - X, y - y, z - Z-, 



^ = —1^-- = -^ = etc. 



Flats with the same orientation may be described as parallel. Parallel 

 flats do not meet, but the converse is not true. 



4. Mutually normal flats. — Two flats Sm and Sm are said to be " normal " 

 or " mutually normal " when each consists of all the right lines which can 

 be drawn through a point perpendicular to all the right lines of the other. 

 If Sm is given, then the orientation of SJ is found by expressing that 



au -r /3/3' + 77' + . . . = 



for all values of the variables A, ju, v, . . . A', m'. "', • • • We have m equations 

 of the type 



I'r Is + 7)l'r r/ls + . . . = 0. 



Hence mf = n - in. Thus the flats normal to an m-flat are (n - m)-Q.a,ts, 

 and one 2^asses through each point of Sm- The two flats intersect in only one 

 point. 



For example, if n = 5, the 3-flat x = 0, y = is normal at the origin 

 to the 2-flat z = 0, u = 0, v = 0. 



5. Co-ordinates of a flat. — Sm is defined by ?i - m linear equations Z = 0, 

 M= 0, N = 0, etc. There are here n (n - m) constants. But we can choose 

 n - m groups of multipliers such that in each group the coefficients of 

 n-m-1 of the variables disappear in \,X + i^rif + v,-N' + . . . Thus 

 Sm is defined by n - m equations, involving constants in number n {n - m) 

 — (n - m - 1). 



Thus the number of independent co-ordinates of an Sm is (n - m]{m + 1). 



* Hence the n co-ordinate axes are mutually perpendicular, 



