SURFACES IN HYPERSPACE. 361 



and 



p + e^ A p + A^ 



uiUi + i\V2 + iviw-i = — Hi^" — h //i" — 2//i — + • — — — = 0. 



These five equations are clearly consistent, H^ — Ei^ +1 = 0, 

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

 Hi first, then Hi and finally H2 and Ho. The sohition is unique — 

 the four apparently different solutions corresponding to changing 

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

 is established as a standard form. 

 The value of h is 2h = {A + (7)ki + Z)ko + /ska , and of 



* = (AC - i^2)k,ki + iC^(kik,> + koki) + |/I£(kik3 + kskO 



+ |Z)£(k2k3+k3k2). 



Here<i>3 = \B-E-D-. If we carry out the linear transformation 



x' = a.r, ij' = ^y, 2/ = 7^1 + bz-i + ezz, %■{ = f ^2 , 23' = ■'?23, 



(124) 

 the surface takes the form 



, 1 l-^A + hP „ 275 , , 7^ + 6/: ,A 



The surface will be unaltered if the relations 



7 = a^. 5 = .la(a - ^)/Z), 6 = C^ - ^)/E, ^ = 0?, V = ^"^ 



are satisfied. There then 0°'^ 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 



7 = a/3 — ' d = a — ; -' e = B > 



B DB ED 



D' F' 



^ 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 



