62 PROCEEDINGS OF THE AMERICAN ACADEMY. 



in the parentheses may be regarded as gotten by operating on the 

 colHneation (dyadic) 



U[ABC]D -[ABD]C+ [B C D] A - [A C D]B\ 



with r, s. For this colHneation the Unear invariant 



^ {{A B C D) - {A B D C) + {B C D A) - (A C D B)\ =0. 



Such a colHneation has sometimes been called normal. By summing 

 we get 



(a r) {b s) — (a s) {b r) = (a r-A s) 



where a ^ is a colHneation such that 



(a A) = 0. 

 Conversely if a ^ is any normal colHneation 



(ar-Ar) = (a A) ^ = 



r being any line or complex. Replacing rhyr-\-s we have 



{a r-A s) -{- (a s-A r) = 0, 



showing that {a r- A s) changes sign with interchange of r and s and 

 is hence of the type 



(a r) (b s) — (a s) {b r). 

 We therefore have 



-=|^> .... (14) 

 (c r) {c s) 



It is to be noticed that this formula determines an angle between two 

 complexes as well as between two lines. In particular the angle is 

 zero if the complexes coincide. 



The system of lines s making a zero angle with a line r = [C D] 

 may be constructed as follows. Let the correspondents of C and D 

 through the colHneation a A be 



C ={Ca)A 

 D' = {D a) A. 



Then s is determined by an equation 



(ar-As) = {a-CD-As) = (a D) {C A s) - (aC) {D A s) 

 = {B' Cs) - {C Bs) ^0. 



