SURFACES IN HYPERSPACE. 311 



Now a^r+i,s+i) = {-ly+s^Ja. 



Hence l/p^ = oErsK+iK+ia^'^'-'"-'^ = aSAr+iX^'+D = a, 



and p = 1/Va. 



Hence the system X^*"^ is 



Further we see easily that 



X, = (-l)'-+i VaXC'+i). (03) 



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^ = {-ly^Ja\r/^la = -K. 



If we have a given system X''"' and let (ps be the covariant system 

 obtained by the composition 



,^, = SrXWX„, (64) 



we have by solution, as may easily be verified, 



Xrs = Xr^s . (64') 



Also <Ps= - ^X'^Xrs , \rs= - \^s ■ (64") 



Thus bv the introduction of tps the svstem Ks of order two is written 

 as the product \r<Ps of two systems of the first order and at the same 

 time Xrs appears as the product — \r(Ps ■ The system cps is called the 

 derived system from the X's 



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

 riant system brs we may form the three invariants, 



a = -Lr.V'^V'^brs. 



|8 = S^XC-^X^fe™. _ (65) 



M = 2„XWX'^^6„ = S,,X('')X^'^&r.. 



