SURFACES IN HYPERSPACE. 



279 



If tlic transformation (3) be effected upon the variables of the Hnear, 



bilinear, , r?i-linear forms, there arise new forms in which 



the coefficients are (Xi), (X,j), . . ., if we now use Ricci's nota- 

 tion,^^ in place of Yi, Ya used above. The law of transformation 

 between the X's and (X)'s is important and is obtained as follows: 



'EiXiXi — ^{Xi^jCijiij — ^i\^iCijXi)y i — Zij{Xj)yj 



Hence 



(Xj) = "LiCijXi or {Xi) = ^jCjiXj. 



If we solve, we have 



Xi = 'Ef/ijiXj). 



Similarly if we take a bilinear form, we find 



"SiijXijXi^j = liijXijLkCikyk^Cjirfi 



= l^ki(^ijCikCjiXij)ykrii = ^ki{Xki)ykrii. 



Hence changing subscripts we have 



{Xij) = 'LkiCkiCijXki, Xij = 'Zfki'Yik'YiiiXki). 



(5) 

 (50 



(6) 



In general for a system of order m, the transformation of the m- 

 linear form shows that the transformation of the system follows the 

 rule 



or 



^imim-^ ilh ' • • im \' ) 



The results of this article may be written more compactly in matrical 

 notation. Let x = (xi, X2, . . . ,.r„), with a similar meaning for y, be an 

 extensive magnitude. Let M be the matrix 



M = 



12 Ricci, Lezioni, p. 49. Although the use of ( ) for the transformed 

 quantities appears awkward it is less so than any notation which has occurred 

 to us. 



