SURFACES IN HYPERSPACE. 365 



which makes dvV = d. If desired it is possible to remove the con- 

 dition that M be a unit tangent plane by writing 



_ n«M d m«M d 



^ ~ W &u W~Fv' 



where M is mxn) We have then, in the general case where n > 3, 



r/M = dT'VM = dr-A, A = VM, (126) 



where M is the unit tangent plane, to correspond to dii = dv^ in 

 the particular case w = 3. 



The dyadic A, however, is one in which the antecedent vectors in 

 the dyads are 1-vectors and the consequent vectors are 2-vectors, 

 i. e., planes, simple or otherwise, — 



A = ii'M — — m'M (12/) 



du dv 



Fiu'ther 



dM dx , 

 —-= — -A = t-A, 

 ds ds 



where t is a unit tangent 1-vector in any direction. The rate of 

 change of the tangent plane in the direction t is therefore t*A. 



The properties of a 1-2 dyadic such as A are not well known and 

 the development of the surface theory from this point of view is there- 

 fore hampered. Some points, however, are readily ascertained. First, 

 there is an invariant or covariant line (1-vector) and an invariant 

 space (3-vector) obtained from A by the familiar processes of insert- 

 ing the signs of scalar and vector products between the elements 

 of the dyadic, — thus 



1 = (n«M)« (m«M)» — 



du dv 



(12S) 



S3=(n.M)x'^-(m.M)x'-^ 



du dv 



By the transformation (b'C)«A = - (C«A)b + C'(bxA), 



1 = — nM« + M« (n X — + mM« M* mX — • 



du \ du / dv \ dv / 



