SURFACES IN HYPERSPACE. 



279 



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



bilinear, , m-linear forms, there arise new forms in which 



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

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

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



^iXiXi = l^iXi'SjCij-yj = ^j{ZiCijXi)y,- = 'S,j{Xj)yj. 



Hence 



(Zy) = ^iCiiXi or (Zi) = S,-c,-iZ,-. 

 If we solve, we have 



Xi = Sy7ij(Z,). 



Similarly if we take a bilinear form, we find 



^ijXijXi^j = liijAijEicCikyk^Cjirji 



= llki(^ijCikCjiXij)ykr}i = ^kiiXkdykVi- 



Hence changing subscripts we have 



(Xij) = -tkiCkiCijXki, Xij = IfkiiikJiiiXki). 



(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 



^imim-^ hh 



Im 



(7) 



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. 



