SURFACES IN HYPERSPACE. 361 



and 



p 4- c~ A P -\- A^ 



uiioi + wi«2 + W1W2 = — ai^' — -, 1- Ih^ — 2//i - + • — — — = 0. 



If Jtf iV 



These five equations are clearly consistent, ILr' — Ai +1 = 0, 

 being redundant. The actual solution could be carried out by finding 

 H\ first, then Si and finally S2 and //o. The solution is unique — 

 the four apparently different solutions corresponding to changing 

 the signs of ii, v, u\ and to interchanging the two sets. Hence (123) 

 is established as a standard form. 

 The value of h is 2h = {A + (7)ki + Dks + A^ks , and of 



$ = {AC - 52)kiki + iCZ) (kiko + koki) + |/I£(kik3 + kgki) 



+ iZ)£(k2k3 + k3k2). 



Here $3 = jB^E^D-. If we carry out the linear transformation 



x' = ax, I/' = I3y, Zi = 7^1 + dzn + ez^ , z^ = fso , 23' = V^s > 



(124) 

 the surface takes the form 



U,^A±8D^ 2yB yC±^ \ ,, ^ ir^,., 



The surface will be unaltered if the relations 



y=a(3, 8 = Aa{a - ^)/D, e = C^ - o.)/E, ^ = a\ r? - ^^ 



are satisfied. There then 00 2 transformations (124) which leave the 

 surface unchanged in the neighborhood of the point 0. Any of the 00 2 

 transformations where 



^ B' , A'&a - B'A^ , C'B& - B'Ca 



D' E' 



^ D ' E 



will carry the surface into one in which the five coefficients are any 

 quantities A', B' , C, D', E'. The determinant of the transformation 



