SURFACES IN HYPERSPACE. 365 



which makes di'V = d. If desired it is possible to remove the con- 

 dition that M be a unit tangent plane by writing 



_ n'M d ni'M d 



^ ~ 'Wdit W~dv' 



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



dM = f/r-VM = f/r-A, A = VM, (126) 



where M is the miit tangent plane, to correspond to dn = dv^ in 

 the particular case n = 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 = n'M m'M (127) 



du dv 



Further 



(/M dr ^ ^ , 

 — - = - -A = t-A, 

 ds ds 



where t is a imit 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 co variant line (1-vector) and an invariant 

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

 ing the signs of scalar and vector products between the elements 

 of the dvadic, — thus 



1 = (ii'M)* (m«M)' — 



dv dv 



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 I 



(128) 



