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 22 = zi{x,y), and any solution Z2 of this equation, 

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

 In case w > 4 we may assume at random w — 3 projections Zi = Zi{x, y), 



zo = Zi{x, y), , Zn-3 = ^n-z{x, 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 tvhich are twisted 

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



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

 we may take 



zi = h{x^ + /), 22 = xy. 



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

 vanish. If we turn the axes of 21 and 32 and of x and y through 45° 

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

 the form 



Zl = ix"^, 22 = h^. 



In this case each of the surfaces 21 = ^x^ and 22 = ^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 



2i = Zi{x), 22 = 22(2/) 



are developable, because the element of arc is 



(1 + 2i'2)c/a;2 + (1 + Z2'^)df = dX' + dY\ 

 dX = Vl + 2i'2 dx, dY ^ <l + Zi'^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-2/-plane. (This is the method followed at length 

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

 for the surface, 



Zi = U^iX^ + ^Bixy + Ca/), i=l,2,...n-2, 



