SURFACES IN HYPERSPACE. 289 



ential or non-euclidean geometry, the angle between the directions 

 bv the formula 



cos e = --"-^^";---^"^ , (25) 



where the X's are proportional to the differentials by (24). The con- 

 dition of orthogonality for the two directions .X^''^ , yX**^ is therefore 





O.^.iXC-'.yXt^) = 0. 



This may be written Ss-iXj-yX^*^ = 0, by using the covariant system. -^^ 

 Our results maj' be simplified by considering the systems i\^^^ , yX^*^ 

 in (24) as first multiplied by such a factor that the radicals in (25) 

 reduce to unity, that is, so that '^rsnrs-i^^''^-i^''^'' = 1. Such a system 

 may be called a unit system. The conditions for a unit orthogonal 

 /(-tuple are therefore, 



2a.iX,.yX(*) = e,y. (26) 



Now if we multii)ly (26) by jX'r , sum over j, and apply (21') w^e have 

 i\r = iK'r- and in like manner we should have ,X('') = i\'^''K Hence 

 for a unit orthogonal /i-tuple the reciprocal and given sets of systems 

 are identical. This gives from (21) the relation 



2:i.iX,.,X(«) = ers , (27) 



in addition to (26) for imit orthogonal n-tuples. The relations (26) 

 and (27) are like those connecting the directions cosines of an orthog- 

 onal set in ordinary space. We may get from (22) the relations 



Si.iX..iX. = ars, 2i..X('-)eX(«) = a^-^ (28) 



14. Transformations of variables. Though the forms in which 

 we are interested are differential and the transformations of variable 

 arbitrary, 



•^■l = ^^l(yi ,y-2,. ■ ., IJn), , •^•« = Xnilll ,ll'l,. . ., Vn), 



18 If we compare this condition of perpendicularity with (22') we see that 

 the direction iX'('") is perpendicular to the direction jX(''^ for all values of i 

 except i = j. If we consider all the directions linearly derived from i\(''), 

 i 9^j, we find that they determine the {n — l)-space perpendicular to j\'('''. 



