362 WILSON AND MOORE. 



is A = a?^^B'D'E' /BDE, and hence the restriction on the quantities 

 is merely that no one of a, /3, B' , D', E' , shall vanish. We see that 

 $'3 ?^ if $3 5^ 0; but that ^z is not an invariant in the ordinary sense 

 of projective geometry that $'3 = A^"$3 — no more is G in the usual 

 surface theory. 



48. Surfaces of revolution. In higher dimensions the simplest 

 type of rotation is that parallel to a plane, all the normals to the 

 plane remaining fixed. If then x = ■i'{s), Zi = Zi{s), i = 1, 2, . . ., 

 be any twisted curve of which s is the arc, a surface of revolution 



X = x{s)cosd, y = .i'(5)sin9, s,- = Zi{s) 



may be obtained by the revolution of the curve parallel to the xy- 

 plane. The surface is made up of circles parallel to the plane with 

 radii equal to the distance of the twisted curve from the z-space of 

 n-2 dimensions. The parameters of the surface are s and 6; the 

 parametric curves are orthogonal. Further 



dy = (i.i-'cos0 + 3y' sind + 'EiiiZi')ds + (- li-sin^ + 3xcoiid)d6. 



m = i.r'cos0 + j.r'sin^ + SkiSi', n = — irsin^ + j.rcos^, 



p = ii-"cos^ + j.r"sin^ + Sk.-Zi", q = — ir'sin^ + j.r'cos^, 



r = — Lrcos^ — j.rsin^. 



As a-'2 + Ss'2 = 1, we have x'x" + Sz'z" = 0, and 



m^ = 1, m-n = 0, n- = .i^ rri'p = 0, m^q = 0, 



m*r = — xx, n'p = 0, n^q = .r.i', n-r = 0, 



flu = 1, «12 = 0, rto'i = -V^, fl = -V", 



Yu = P, yi2 = q - .i-'m/x = 0, y^o = r + xx'm. 



The element of arc is ds^ -\- x'^dB^. It therefore appears that: The 

 surface of revolution is always applicable upon a surface of revolution 

 in three dimensions in which the directrix in the .r«/-plane is a; = x{s), 

 z = z(s). [The equation z = z(s) is redundant and so is one of the 

 71 — 2 equations z; = Zi{s)]. ' ' ' 



The value of G is q-r/a = — x"/x. The condition 6' = for a 

 developable is therefore x" = or x = CiS + co which establishes 

 between the differentials the relation dx = cids or 



(1 - Ci')dx^ = Ci^dZi" + r/so2 4- . . + dZn-'r), Ci < 1. 



