342 WILSON AND MOORE. 



we assume the projection Zi = Zi{x, y) at random, (108') equated to 

 zero becomes a partial differential equation of the second order for 

 the other projection zo = Z2{x,y), and any solution Zo of this equation, 

 taken with Si, will define a developable surface in four dimensions. 

 In case w > 4 we may assume at random n — 3 projections Zi = 2i(x, y), 



z-2 = z^ix, y), , z„_3 = Zn-zix, y), and proceed to solve the 



differential equation obtained by setting (108) equal to zero for the 

 projection z„_2 which taken with the assumed n — 3 projections, will 

 determine a developable. In more than three dimensions developable 

 2-surfaces therefore are either 1°, ruled developables which are twisted 

 curve surfaces, or 2°, non-ruled developable surfaces. 



As a particularly simple case of a non-ruled developable for w = 4 

 we may take 



zi = i(.r- + r)» 22 = xy. 



This surface satisfies (108') but the individual terms rt — s- do not 

 vanish. If we turn the axes of Zi and Z2 and of x and y through 45° 

 in their respective planes and change the scale, the surface may take 

 the form 



Zl = |.t2, Z2 = ^/. 



In this case each of the surfaces zi = ^.t^ and Z2 = ^y^ taken as a three 

 dimensional surface is developable. But the four dimensional 

 surface is not a ruled surface. In other words the projections of a 

 non-ruled developable may each be ruled developables. All surfaces 

 of the type 



si = zi(x), Z2 = zoiy) 

 are developable, because the element of arc is 



(1 + zi'2)rfa;2 + (1 + Z2'^)df = dX^ + dY\ 

 dX = Vl + zi'2 dx, dY = Vl + 22'2 dy. 



Such surfaces, however, are not in general ruled. 



42. Development of a surface about a point. There is a great 

 simplification in our formulas if we restrict ourselves to the neighbor- 

 hood of a single point of the surface and take the tangent plane at 

 that point as the a-^-plane. (This is the method followed at length 

 by Kommerell in the four dimensional case.) In general we have 

 for the surface, 



Zi = A(yl,r2 + 2B,xy + C/), i= 1,2,.. .n - 2, 



