SURFACES IN HYPERSPACE. 367 



One of the first problems in discussing 1-2 dyadics of the type 

 aA + bB would be the establishment of a standard form. If a and b 

 be replaced by linear combinations xdJ + yh' , x'dJ + y'h' of two 

 vectors in their plane, the dyadic becomes 



A = a'(xA + a-'B) + h'{yk + y"B) = a'A' + b'B', 



where A', B' are linear combinations of A and B. We may then 

 consider that for the antecedents a', b' we have chosen unit normal 

 vectors i, j so that A = iA + JB. If a rotation is carried out on i, 

 j, we have 



A = i'(Acos^ - B sin^) + j'(A sin.^ + B cos^) = i'A' + J'B'. 



The condition A'«B' = 0, i. e., the condition that the consequents be 

 orthogonal is that ^ be determined from 



tan2^ = 2A-B/(A2 - B2), 



which gives four values of (p spaced at right angles. Hence: We may 

 reduce A to the form 



A=iA + jB, A-B = 0, (129) 



and this reduction is unique (except for the indeterminateness of an 

 interchange of i and j or a reversal of the sign of either). The reduc- 

 tion is wholly indeterminate when ip is indeterminate, i. e., when A«B 

 = and A- = B^. In the special reduced form (129), the directions 

 i, j are along the principal directions on the surface in case we use for 

 principal directions the definition 3 introduced and preferred by tis.^^ 

 If j is chosen relative to i and M so that M = ixj, we have 



a = i«A«j, P = — j*A«i, K- = — i«A«i = j»A»j. 



As ij — ji is a dyadic independent of the directions of i in M, 



2h= (ij-ji):A = a+p 



50 Note the correspondence with three dimensions. If we have the dyadic 

 aa + bp = 4> where dn = dr.*, we may reduce to the form ia + jp, 

 a.p = 0, which is a reduction to the principal axes of "t>, and have then i, j 

 along the principal directions, since * is self-conjugate. 



