SURFACES IN HYPERSPACE. 343 



up to infinitesimals of the third order. The 3/i — G constants are not 

 geometrically independent because of the arbitrary choice of the 

 direction of the axes in the .r^-plane. There are only 3n — 7 independ- 

 ent constants. There are 3(w — 2) — 6 = 3n — 12 degrees of freedom 

 for a plane in Sns and five degrees of freedom for an ellipse in the 

 plane. The count of constants indicates, therefore, that the indicatrix 

 may be any ellipse in the normal space. We may examine this 

 proposition critically bv reference to the general formulas (107), 

 (1070, (107"). 

 For this case m = i, n = j, an = 1, ai2 = 0, 022 = 1, a = 1. 



a(u) = 1, a('2' = 0, aP2) = i, p = ^A^ki, q = ^BMi, 

 r = SCiki, yn = p, yi2 = q, y22 = r, 2h = S(^i + Ci)ki. 



The center of the indicatrix may therefore be any point, and is the 

 same point for any two surfaces for which Ai-{- d are the same, 

 i = 1, 2,. . .n — 2. The plane K-^S is determined by (102) as 



2|lx8 = yi.x(y2o - yn) = S5,kixV(C,. - Aj)k,: 



As Aj -j- Cj and Aj — C, are independent, |Xx8 may be any plane in 

 the normal /Sn_2. The work may now be simplified by choosing h 

 as the axis Zi and by taking the axes Zo, 23 in the space hxfJ^xS. The 

 equations of the surface reduce to 



zi = M^i^' + 2Bi.n/ + C,f], Z2 = i[^2(a:2 - if) + 2B,.nj], 

 23 = M^3(.f- - if) + 2Bsxij], Zi =0, i = 4,5,...n-2. 



That 22 and 23 take these special forms is due to the fact (§38) that the 

 mean curvature of each must vanish. A proper orientation of the xy 

 axes makes Bi = 0. By properly choosing the axes k2, ks we may 

 make B2 vanish. We have then as a canonical form for the surface, 



21 = ^[A.x-^ + C\y'], 22 = hM--c' - y~), ' (109) 



23 = WAzix' - f) + 2Bzxy], Zi =0, i> 3. 



Now, 2h = (vli + r,)ki , 2|ix5 = 53k3x[(Ci - ^i)ki - 2^2ko]. 

 If we set Ai = /? + r, (\ = h — e, A-i = f, A3 = A, B3 = B, 

 21 = m^' + !f) + eix'^ - y')], 22 = i/(x2 - f), 



23 = hUix- - y~) + 2Bxy], Zi =0, i > 3. (109') 



