344 
ON THE 34 CONCOMITANTS OF THE TERNARY CUBIC. 
[770 
Each concomitant of the general form is an indecomposable function, not breaking 
up into rational factors; but this is not of necessity the case in regard to a canonical 
form: only a concomitant which does break up must be regarded as indecomposable, 
no factor of such concomitant being rejected, or separated. So far from it, there is, 
in regard to the canonical form in question, a frequent occurrence of abc + 8I s or a 
power thereof, either as a factor of a unique concomitant, or when there are two 
or more concomitants of the same deg-class-order, then as a factor of a properly 
selected linear combination of such concomitants: and the principle referred to is, in 
fact, that of the selection of such combination for the representative concomitant; or 
(in other words) the representative concomitant is taken so as to contain as a factor 
the highest power that may be of abc + 8l 3 . As to the signification of this expression 
abc + 81 s , I call to mind that the discriminant of the form is abc (abc + 81 3 ) 3 . 
As to numerical factors: my principle has been, and is, to throw out any common 
numerical divisor of all the terms: thus I write S = — abcl + l 4 , instead of Aronhold’s 
S = — 4abcl + 4Z 4 . There is also the question of nomenclature: I retain that of my 
Seventh Memoir on Quantics, except that I use single letters H, P, &c., instead of 
the same letters with U, thus HU, PU, &c.; in particular, I use U, H, P, Q 
instead of Aronhold’s f, A, 8/, Tf. It is thus at all events necessary to make some 
change in Gundelfinger’s letters; and there is moreover a laxity in his use of accented 
letters; his B, B', B", B!", and so in other cases E, E', E", &c., are used to denote 
functions derived in a determinate manner each from the preceding one (by the 
3-process explained further on); whereas his L, E; M, M'; N, N' are functions 
having to each other an altogether different relation; also three of his functions are 
not denoted by any letters at all. Under the circumstances, I retain only a few of 
his letters; use the accent where it denotes the 3-process; and introduce barred 
letters J, K, &c., to denote a different correspondence with the unbarred letters J, 
K, &c. But I attach also to each concomitant a numerical symbol showing its 
deg-class-order, thus: 541 (degree = 5, class = 4, order = 1) or 1290, (there is no 
ambiguity in the two-digit numbers 10, 11, 12 which present themselves in the system 
of the 34 symbols); and it seems to me very desirable that the significations of 
these deg-class-order symbols should be considered as permanent and unalterable. 
Thus, in writing S = 400 = — abcl +1 4 , I wish the 400 to be regarded as denoting its 
expressed value — abcl +l i \ if the same letter S is to be used in Aronhold’s sense 
to denote — 4<abcl + 4Z 4 , this would be completely expressed by the new definition 
S = 4.400, the meaning of the symbol 400 being explained by reference to the present 
memoir, or by the actual quotation 400 = — abcl + l i . 
I proceed at once to the table: for shortness, I omit, in general, terms which 
can be derived from an expressed term by mere cyclical interchanges of the letters 
(a, b, c), (f, v, ( x > V> 2 )•