SURFACES IN HYPERSPACE. 331 



in conformity wdth the expression for the analogous quantity in three 

 dimensions. 



The plane 5x|ji of the indicatrix is polar to h with respect to Cone I, 

 that is, it is perpendicular to 4>-i*h, as may also be seen by direct 

 substitution in (94'). 



The second fundamental forms \pi of the projection of the surface 

 on any 3-space containing the tangent plane M at a point and some 

 normal Zjisi/', = Zj*^. If we write, 



^ = ZZiii = ZZiZi-^F = 1-^, 



the mean curvature of the surface is seen to be the vector sum of the mean 

 curvatures of the lirojections on the spaces determined successively by Zj, 

 namely, 



2h = ^ijU^^'^Yij = Ziju{a^''>yirZk)Zk. 



There is no need of letting k vary over more than the values 1, 2, 3, 

 as curvature phenomena are five dimensional. The expression for G 

 may be written 



aG = fis = ^:I = ^:l = yii*I-y22 — yi2*I*y2i 



= 2fc(yii'ZfcZfc'y22 - yi2-ZiZfc-y2i). 



As the individual parentheses here are the \'alues of G for the projec- 

 tions of the surface it shows that: The total curvature of a surface is the 

 algebraic sum of the total curvatures of the orthogonal projections of the 

 surface. 



Since aG = 4>:(ZiZi + z^Zo + Z3Z3) we may reduce aG to a single 

 term by choosing Z2 and Z3 on the cone r*$'r — 0, i. e., upon Cone II. 

 Then aG = Zi*$*Zi. As aG = $s, this relation may be written as, 



$sZi'Zi = Zi'$'Zi 



or Zi'($sl — 4>)'Zi = 0. 



We therefore have another cone, 



r- ($sl - *) T = 0, Cone III, (96) 



which is coaxial with Cones I and II and which has the property that 

 if one normal Zj lies upon it, the other two may lie upon Cone II, and 



