He laid down the following arbitrary rules for determining the number of atoms in a compound: - if only one compound of two elements exists, it is a binary compound and its atom is composed of one atom of each element; if two compounds exist one is binary (say A + B) and the other ternary (say A + 2B); if three, then one is binary and the others may be ternary (A ± 2B, and 2A + B), and so on.
If a compound contains two atoms it is termed a binary compound, if three a ternary, if four a quaternary, and so on.
This view was accepted in 1817 by Leopold Gmelin, who, in his Handbuch der Chemie, regarded inorganic compounds as being of binary composition (the simplest being oxides, both acid and basic, which by combination form salts also of binary form), and organic compounds as ternary, i.e.
The idea_can be generalized so as to have regard to ternary and higher forms each of the same order and of the same number of variables.
Ternary and Higher Forms.-The ternary form of order n is represented symbolically by (aixl+a2x2+a3x3)' =a'; and, as usual, b, c, d,...
For example, take the ternary quadratic (aixl+a2x2+a3x3) 2 =a2x, or in real form axi +bx2+cx3+2fx 2 x 3+ 2gx 3 x 1 +2hx i x 2.
The ternary cubic has been investigated by Cayley, Aronhold, Hermite, Brioschi and Gordan.
Hesse showed independently that the general ternary cubic can be reduced, by linear transformation, to the form x3+y3+z3+ 6mxyz, a form which involves 9 independent constants, as should be the case; it must, however, be remarked that the counting of constants is not a sure guide to the existence of a conjectured canonical form.
Thus the ternary quartic is not, in general, expressible as a sum of five 4th powers as the counting of constants might have led one to expect, a theorem due to Sylvester.
This is of degree 8 in the coefficients, and degree 6 in the variables, and, for the canonical form, has the expression -9m 6 (x 3 +y 3 +z 3) 2 - (2m +5m 4 +20m 7) (x3 +y3+z3)xyz - (15m 2 +78m 5 -12m 8) Passing on to the ternary quartic we find that the number of ground forms is apparently very great.
The system of two ternary quadratics consists of 20 forms; it has been investigated by Gordan (Clebsch-Lindemann's Vorlesungen i.
Ciamberlini has found a system of 127 forms appertaining to three ternary quadratics (Batt.
For the unipartite ternary quantic of order n he finds that the fundamental system contains a (n+4) (n -1) individuals.
He successfully considers the systems of two and three simultaneous ternary quadratics.
Of the Memoir he discusses bi-ternary quantics, and in particular those which are lineo-linear, quadrato-linear, cubo-linear, quadrato-quadratic, cubo-cubic, and the system of two lineo-linear quantics.
He shows that the system of the bi-ternary n°m i ° comprises 4 (n+1)(n+2)(m+1)(m+2)- 3 individuals.
Bibliographical references to ternary forms are given by Forsyth (Amer.
For instance, those of a ternary form involve two classes which may be geometrically interpreted as point and line co-ordinates in a plane; those of a quaternary form involve three classes which may be geometrically interpreted as point, line and plane coordinates in space.
The ternary alloys containing bismuth, tin and lead have been studied in this way by F.
The alloy of the point e is the ternary eutectic; it deposits the three metals simultaneously during the whole period of its solidfication and solidifies at a constant temperature.
It is evident that any other property can be represented by similar diagrams. For example, we can construct the curve of conductivity of alloys of two metals or the surface of conductivity of ternary alloys, and so on for any measurable property.
This subject is far from being exhausted, and it is not improbable that the alloy-producing capacity of aluminium may eventually prove its most valuable characteristic. In the meantime, ternary light alloys appear the most satisfactory, and tungsten and copper, or tungsten and nickel, seem to be the best substances to add.
The British method is a mixture of the last two, but with an index-scale which is partly ternary and partly binary.
We have in the Hessian the first instance of a covariant of a ternary form.
The theory of the invariants and covariants of a ternary cubic function u has been studied in detail, and brought into connexion with the cubic curve u = o; but the theory of the invariants and covariants for the next succeeding case, the ternary quartic function, is still very incomplete.