SURFACES IN IIYPERSPACE. 289 



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

 by the formula 



COS 6 = — — — ■ — , ^-^^^^ , K^o) 



V"^ « \{r) \U) \V „ .\{r) .\(s) 



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

 dition of orthogonality for the two directions iX*""' , jV'> is therefore 



This may be written Ss.iXs.yX^'^ = 0, by using the covariant system. ^^ 

 Our results may be simplified by considering the systems iX^'"^ , ,X(^' 

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

 reduce to unity, that is, so that Sr8ars-iX"^'.,:X''^ = 1- Such a system 

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

 n-tuple are therefore, 



2,.iX..,X(^) = 6.y. (26) 



Now if we multiply (26) by yX'r , sum over j, and apply (21') we have 

 ,.X^ = iX'r. and in like manner we should have iX''' = iX'^'K Hence 

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

 are identical. This gives from (21) the relation 



2i.iX..,-X(«) = 6., , (27) 



in addition to (26) for unit orthogonal /i-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 



14. Transformations of variables. Though the forms in which 

 we are interested are differential and the transformations of variable 

 arbitrary, 



X'l = .Vl(^l ,lj2.,. ■ ■, IJn), , -IV = '^n{yi ,y-l,. ■ ., l/n). 



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

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

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

 i ^j, we find that they determine the {n - l)-space perpendicular to iX'^'l 



