SURFACES IN HYPERSPACE. 363 



In case n = 3 the solution is immediate, viz. z = nix -\- b, a line. In 

 case 71 > 3 we may assign to w — 3 of the variables Zi any arbitrary 

 values as functions of x (provided that the sum of dz^ is not too large 

 if we desire a real surface). For instance if we consider the case 

 n = 4 and let Zi = aicosbx, 



(1 — Ci^ — Ci^ai'^bh\n'^bx)dx" = c^dz^, or (1 — c^)co&%xdx'^ = c^dz^ 



if we choose c^a^b"^ = (1 — c^) to simplify the integration for a 

 particular case. Then 



Vl — c^ 



22 = ; sinfea- + C = ai^mbx -\- C. 



Cib 



The curve Zi = aicosbx, Z2 = aisin6.r is a circular helix about the 

 axis of X in the xziZ2 space. The four dimensional surface of revolu- 

 tion is 



2l 



= aicosfe "Va;^ + y^, zi = aisinfe ^x^ + y^. 



We see therefore that: The developables of revolution when n > 3 form 

 an extended class of surfaces instead of reducing merely to the cones and 

 cylinders. 



The value of h is given by 



2h = p + (r + xx'tdl)/x^. 



If we designate i cos0 + j sin0 by u, a unit vector, 



p = x"n + Ski^i", r = — .ru, m = x'n + Skiz/ 

 and 



2h = m{xx" + x'^)/x + Ski(z/' + x'z'/x). 



The condition h = for a minimal surface therefore is 



XX" - 1 + .C'2 = 0, Zi" + x'z^x = 0. 



The last equation shows that ZiX = Ci, the first that x^ = (s -\- by 

 4" a^. Hence 



= cosrt"^ - , I = 1, 2,. . ., n — 2, 



d a 



