362 WILSON AND MOORE. 



is A = o?fi^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 ?^ 0; but that <l>3 is not an invariant in the ordinary sense 

 of projective geometry that $'3 = A'^^s — 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 = x{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 = x{s)^m.d, Zi = 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 s-space of 

 71-2 dimensions. The parameters of the surface are s and 6; the 

 parametric curves are orthogonal. Further 



dy = (u''cos0 + j^'sin^ + Ski2/)f/i^ + (— irsin^ + j.rcos0)c?5. 



m = u-'cos0 + ix'&md + SkiZi', n = — i.rsin0 + j.rcos^, 



p = ix"cos0 + j.r"sin0 + Skiz/', q = — i.r'sin0 + j.r'cos^, 



r = — i.rcos0 — j.i'sin0. 



As .r'2 + Sz'2 = 1, we have x'x" + Sz's" = 0, and 



m^ = 1, Hi'ii = 0, n^ = .1", ni'P = 0, in*q = 0, 



niT = — xx', n-p = 0, n«q = xx' , iit = 0, 



Oil = 1, 012 = 0, a22 = x"^, a = x"^, 



yii = P, yr2 = q - x'm./x = 0, y22 = r + .r.i-'m. 



The element of arc is ds^ + x'dd". 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 x = x{s), 

 z = z{s). [The equation 2 = z{s) is redundant and so is one of the 

 71 — 2 equations Z; = Zi(s)]. 



The value of G is qT/a = — x"/x. The condition G^ = for a 

 developable is therefore x" = or x = CiS + c-i which establishes 

 between the differentials the relation dx = Cids or 



(1 - Ci')dx^ = Ci2(ffei2 + ffe.- + . . + dZn--:?), Ci < 1. 



