143 



mental eontravariants PU, QU do in my third memoir ; in the nota- 

 tion of the present memoir these derived eontravariants are 



YU=3T. PU-4S.QU, 

 ZU=-48S 2 .PU+T.QU; 



and for the canonical form x 3 +y 3 +z 3 -\-6lxyz, they acquire respect- 

 ively the factor (1 + 8 3 ) 2 , viz. in this case 



YU=(1+8OM* 



The derived eontravariants have with the covariants U, HU, even a 

 more intimate connexion than have the eontravariants PU, QU ; and 

 the advantage of the employment of YU, ZU fully appears hy M. 

 Aronhold's memoir. 



But the conclusion is, not that the eontravariants PU, QU are to be 

 rejected, but that the system is to be completed by the addition thereto 

 of two derived covariants, linear functions of U, HU ; these derived 

 covariants, suggested to me by M. Aronhold's memoir, are in the 

 present memoir called CU, DU ; their values are 



CU=-T . U + 24S.HU 

 DU= 8S 2 .U-3T.HU: 



and for the canonical form x 3 +y a -}-z 3 + 6lxyz, they acquire respect- 

 ively, not indeed (1 -f 8J 3 ) 2 , but the simple power (1 + 8/ 3 ), as a factor, 

 viz. in this case 



it was in fact by means of this condition as to the factor 

 that the foregoing expressions for CU, DU were obtained. 



The formulae of rny third memoir and those of M. Aronhold are 

 by this means brought into harmony arid made parts of a whole ; 

 instead of the two intermediates aU + 6/3HU, 6aPU+/3QU in Tables 

 68 and 69 of my Third Memoir, or of the intermediates oU+6/3HU 

 2aYU + 2/3ZU of M. Aronhold's theory, we have the four interme- 

 diatesaU + 6/3HU,~-2aYU+2/3ZU,2aCU-2/3DU,6aPU+/3QUin 

 Tables 74, 75, 76, and 77 of the present memoir. These four tables 

 embrace the former results, and the new ones which relate to the co- 

 variants CU, DU ; and they are what is most important in the pre- 



