SURFACES IN HYPERSPACE. 331 



in conformity \\'ith the expression for the analogous quantity in three 

 dimensions. 



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

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

 substitution in (94')- 



The second fundamental forms i/', of the projection of the surface 

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

 normal z,isi/', = z^*^. If we write, 



^ = 2z,i/', = :i:z,z,'"^ = I'^P, 



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

 curvatures of the jyrojections on the spaces determined successively by z„ 

 namely, 



2h = S,ya(''Vi; = 2i,fc(a(^')yiyZit)Zfc. 



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 = 9.S = 9.:l = $:I = yii*I*y22 — yi2*I'y2i 



= 2fc(yii-Zi.Zfc-y22 - yi2-ZfcZ;fy2i). 



As the individual parentheses here are the values 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 yrojections of the 

 surface. 



Since aG = <i>:(ziZi + Z2Z2 + Z3Z3) we may reduce aG to a single 

 term by choosing Z2 and Z3 on the cone T'^'X = 0, i. e., upon Cone 11. 

 Then aG = Zi«$«Zi. As aG = $sj this relation may be written as, 



$sZi*Zi = Zi*$*Zi 



or Zi*(<S>sI — $)*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 



