SURFACES IN HYPERSPACE. 311 



Now a^'^+h^+i) = {-ly+'Urs/a. 



Hence l/p^ = aS,>,+iX,+ia(^+i- '+i) = aSAr+iX''"+') = a, 



and p = 1/Va. 



Hence the system X^*"^ is 



Further we see easily that 



X. = (-1)'-+^ VaX('-+i). (63) 



The system X^''^ or Xr is called the canonical orthogonal system for 

 X^*") or Xr . The repetition of the process of forming the canonical sys- 

 tem leads to the negative of the original system (not to the system 

 itself). For 



X, = (-!)'• +WaX(^+i^ = (-l)NaX,/Va = - X^. 



If we have a given system X^""' and let cps be the covariant system 

 obtained by the composition 



<p, = ZrXO-^X,,, (64) 



we have by solution, as may easily be verified, 



y<rs = 'Krfs • (64 ) 



Also cps= -'Er\^'^\rs, \rs= - K<Ps ■ (64") 



Thus by the introduction of (pg the system \rs of order two is written 

 as the product Xrfs of two systems of the first order and at the same 

 time Xrg appears as the product — \r^s • The system 953 is called the 

 derived system from the X's. 



27. Expressions of the second forms. If we consider a cova- 

 riant system bra we may form the three invariants, 



/3 = 2;„X(^^X(^)6„. (65) 



u = S„XWX'-'^&„ = S„X('-)X(«^6„. 



