Theory of Canonical Forms and of HyperdeterminanU, 407 



The reduction of the function of the fourth degree to its ca- 

 nonical form may be effected very easily by means of the proper- 

 ties of the invariants of the canonical form, as I have shown in 

 the paper in the Cambridge and -Dublin Mathematical Journal 

 before alluded to. Accordingly I have endeavoured to ascertain 

 whether the reduction of the .sixth degree might not be ejBfected 

 by a similar method. ' oil j lo 



If we start with the ibvm' aa^ + bif -^ c^ ■\-^Omx^y'^z^, where 

 x-\-y-\-z^=0, and which is only another mode of representing 

 the canonical form previously given, we shall find that there are 

 four independent invariants of the second, fourth, sixth, and 

 tenth degrees. Calling these Hg, H4, Hg, H^q, and writing 

 ^}y ^2> ^3 fo^' a + b + c,ab + ac + bc,abc, it will be found, after 

 performing some extremely elaborate computations, that ^'^^'i^ 

 , ... Ec,=s^-270m^ .-i.^-..\:r >ivi)^Tl 



H4 = 6ms3 + 4^6m\ + 2l6m\ + SOlm^ '^olI(a »4:f ni^jdo I 

 H6=4^3Hl20v3^-{68452^ + 4325i53}m^ ^^-^^^^.^S^ 

 + (13 . 27 . 6453-64 . SlsiSc,)m^ -{-S . 81 . 1695^ 

 + 7 . 128 . 729^1 . m^ + 16 . 729 . 239m«. 



Hjo is too enormously long to attempt to compute ; but we 

 can easily prove its independent existence by making m = 0, in 

 which case the (determinant, or, to use the new term proposed, 

 the) discriminant of aw^ + by^ + cs^ becomes the product of the 

 twenty-five forms of the expression 



{abf-\-{acf.r+{bcf.r^. 



Now in general the value of such a product for a^+yS^l -f 7 . 1 

 is obviously of the form 



(« + /3 + 7)^ + a^7(/a + /e + 72+^ ajS + ay -{- fiy) ; ihny 



for when a=0 or ;S=0 or 7=0, the product must become re- 

 spectively (/3-{-7)^, (7 + a)^, and (a + /3)^. Moreover, without 



* Such a product in the language of the most modern continental ana- 

 lysis is, I believe, termed a Norrae. If we suppose the general function 

 of w, y of the 4th degree thrown under the form A%^4-Bz>^+Cw^ where 

 u+v-\-w=0, and the general function of x, y, z of the 3rd degree thrown 

 under the form Aw^+Bzj^+Ciy^+D^^, where u-\-v-\-w-\-6=zQ,ihQ theory 

 of noraies will afford an instantaneous and, so to speak, intuitive demon- 

 stration of the respective related theorems, that the discriminant {aliter 

 determinant) of each such function is decomposable into the sum of a 

 square and a cube. Each of these forms is indeterminate, in either case 

 there being but two relations fixed between the coefficients A, B, C ; A, 

 B, C, D ; and we may easily establish the following singular sgecies of alge- 

 braical porism. In the first case .^3 b^qoL i^™ iisstf 



(ABC)2 : (AB+AC+BC)3, Ja«w fjtli rao-d -gukm 

 and in the second case ? isdi b9iJ[D«9i Trad 



(ABCD)3 : (2A2B2C2-2ABCD2AB)2 ,*M 



are invariable ratios. 



