Theory of Canonical Forms and of Hyper determinants, 397 



and 



(x + \y){x-\-\^){x-\-\^)[x^X^y) 



we shall then have nine equations for determining the nine 

 unknown quantities of the general forni 



p{K[ +p^'^ +;?3\* +p^\ + M,m = a,y 

 where l has all values from to 8 inclusive/ and where 

 ^ (1.2...)(l.2...(8-0) 



1.2. ..8 

 multiplied into the coefficient of y'^ , x^~^ in U^. 



Taking these nine equations in consecutive fives,, beginning 

 with the first, second, third, fourth, fifth, and ending with the 

 fifth, sixth, seventh, eighth, ninth, we obtain the five equations 

 following : — 



«o • s^--a^s^ + ttc^Sc^^a^ .s^ + a^, ^q— mNi=:0 

 «i . 54— a^s^ + a^s^—a^^ . 5, + a^ . 5q— mN2=0 

 «2 . s^—(IqSq + a^s^—a^ .s^ + aQ. Sq— mN3=0 

 % . 54— ^453 + CgSg— ffg . 5^ -f «7 . ^Q— mN4=0 



«4 . s^—a^s^+af^s^—aj .s^+aQ. SQ—mlSi^=0, 

 where 



Ni=MoS4-Mi . 53 + M2S2-M3 . Si + M4 



N2 = Mi54-M2.53 + M3S2-M4.5i + M5 

 N3 = M2S4-M3 . S3 + M452-M5 .Si+Mq 



N4=M3S4— M4 . ^g + Mg^g—^e • ^1 + M7 



/ .' N5 = M454-M5.53 + M652-M7.5l + M8. 



Developing now XJ^, we obtain 



Mo=70 Mi=^., M2=5.2+|.,^ M3=|.3+|v. 



M4= 2^4 + 2^1^ + 522 M5=^5i54+-^2-% M6= 5^2-^4 +1^3' 



Hence Nj = 72^4-185153 + G^g^ 



N2=185i54-^V%+-522 

 N3 = 1 252^4 — 351^2 • h + ^2^ 



9 3 



N4= I85354- ^ 51532+ g . ^2^ 53 

 N5= 72542 - 185^53 . 54 + 652^ . 54. 



P/«7. ilf«^. S. 4. Vol. 2. No. 12. Nov. 1851. 2 E 



