SURFACES IN HYPERSPACE. 363 



In case ?i = 3 the solution is immediate, viz. z = mx + 6, a line. In 

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

 values as functions of x (provided that the sura 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%h\Ti^bx)dx^ = Ci^dz^^, or (1 — C]^)cos'^bxdx'^ = Cx-dz-? 



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

 particular case. Then 



Vl — ci2 



22 = — svabx -{■ C = ai^inhx + C. 



Cib 



The curve Zi = aiCos6.r, 22 = aisin6.r is a circular helix about the 

 axis of X in the a:2i22 space. The four dimensional surface of revolu- 

 tion is 



2i = aiCos6 "Vx- + 2/^ 22 = aisinft > 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'm)/x\ 



If we designate i cos9 + j sin^ by u, a unit vector, 



p = x"\i + Skis/', r = — xu, m = x'u + Skis/ 

 and 



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



The condition h = for a minimal surface therefore is 



xx" - 1 + x'2 = 0, , Zi" + xW/x = 0. 



The last equation shows that z/x = d, the first that x^ = (5 + by 

 + a^. Hence 



?ii-^=cosA-i"^, z= l,2,...,n-2, 

 Ci a 



