The name lemniscate is sometimes given to any crunodal quartic curve having only one real finite branch which is symmetric about the axis.
The criterion whether a pseudo-symmetric form is a true polymorph or not consists in the determination of the scalar properties (e.g.
Although many pseudo-symmetric twins are transformable into the simpler form, yet, in some cases, a true polymorph results, the change being indicated, as before, by alterations in scalar (as well as vector) properties.
The theories of determinants and of symmetric functions and of the algebra of differential operations have an important bearing upon this comparatively new branch of mathematics.
+amam Expanding the right-hand side by the exponential theorem, and then expressing the symmetric functions of al, a2, ...a m, which arise, in terms of b1, b2, ...'
B., we obtain by comparison with the middle series the symbolical representation of all symmetric functions in brackets () appertaining to the quantities p i, P2, P3,ï¿½ï¿½ï¿½ To obtain particular theorems the quantities a l, a 2, a 3, ...a, n are auxiliaries which are at our entire disposal.
When a skew symmetric determinant is of even degree it is a perfect square.
A skew determinant is one which is skew symmetric in all respects,.
If al, a2, ...a, n be the roots of f=o, (1, R2, -Ai the roots of 0=o, the condition that some root of 0 =o may qq cause f to vanish is clearly R s, 5 =f (01)f (N2) ï¿½ ï¿½;f (Nn) = 0; so that Rf,q5 is the resultant of f and and expressed as a function of the roots, it is of degree m in each root 13, and of degree n in each root a, and also a symmetric function alike of the roots a and of the roots 1 3; hence, expressed in terms of the coefficients, it is homogeneous and of degree n in the coefficients of f, and homogeneous and of degree m in the coefficients of 4..
There is no difficulty in expressing the resultant by the method of symmetric functions.
Now by the theory of symmetric functions, any symmetric functions of the mn values which satisfy the two equations, can be expressed in terms of the coefficient of those equations.
THE Theory Of Symmetric Functions Consider n quantities a l, a 21 a 3, ...
N be permuted, is a rational integral symmetric function of the quantities.
+ax n, al, a2, ...an are called the elementary symmetric functions.
A separation is the symbolic representation of a product of monomial symmetric functions.
) j1+j2+j3+..ï¿½ (J1+ j2 +j3+...-1)!/T1)?1(J2)72 (J 3)/3..., j11j2!j3!... ?.1 for the expression of Za n in terms of products of symmetric functions symbolized by separations of (n 1 1n 2 2n 3 3) Let (n) a, (n) x, (n) X denote the sums of the n th powers of quantities whose elementary symmetric functions are a l, a 2, a31ï¿½ï¿½ï¿½; x 1, x2, x31..; X1, X2, X3,...
In terms of x 1, x2, x3,ï¿½ï¿½ The inverse question is the expression of any monomial symmetric function by means of the power functions (r) = sr. Theorem of Reciprocity.-If ï¿½1 P2 "3 01 Q 2 7 3 Al A 2 A3 X m1 X m2 X m3 ...
The general monomial symmetric function is a P1 a P2 a P3.
(0B) = (e), &c. The binomial coefficients appear, in fact, as symmetric functions, and this is frequently of importance.
The sum of the monomial functions of a given weight is called the homogeneous-product-sum or complete symmetric function of that weight; it is denoted by h.; it is connected with the elementary functions by the formula 1 7r1l7r2!7r3!
- " If a symmetric function be symboilized by (Aï¿½v...) and (X1X2X3..ï¿½), (ï¿½i/-12ï¿½3ï¿½ï¿½ï¿½), (v1v2v3...)...
The law of reciprocity shows that p(s) = zti (m 1te2tmtL3t) t=1 st It 2t 3t viz.: a linear function of symmetric functions symbolized by the k specifications; and that () St =ti ts.
" The symmetric function (m ï¿½8 m' 2s m ï¿½3s ...) whose is 2s 3s partition is a specification of a separation of the function symbolized by (li'l2 2 l3 3 ...) is expressible as a linear function of symmetric functions symbolized by separations of (li 1 12 2 13 3 ...) and a symmetrical table may be thus formed."
The introduction of the quantity p converts the symmetric function 1 2 3 into (XiX2X3+...) -Hu Al (X 2 A 3 .-) +/l02(X1X3.ï¿½.) +/103(A1X2.ï¿½.) +....
P operators D upon a monomial symmetric function is clear.
It has been shown (vide " Memoir on Symmetric Functions of the Roots of Systems of Equations," Phil.
A Product of Symmetric Functions.
- Suppose f to be a product of symmetric functions f i f 2 ...f m .
Application to Symmetric Function Multiplication.-An example will explain this.
Which is satisfied by every symmetric fraction whose partition contains no unit (called by Cayley non-unitary symmetric functions), is of particular importance in algebraic theories.
For example, the theory of invariants may be regarded as depending upon the consideration of the symmetric functions of the differences of the roots of the equation aox n - (i) a i x n - 1 + (z) a 2 x n 2 - ...
The important result is that the theory of invariants is from a certain point of view coincident with the theory of non-unitary symmetric functions.
=o, are symmetric functions of differences of the roots of aox n - 1!(n)a4xn-1+2!()a2xn-2-...
= 0; and on the other hand that symmetric functions of the differences of the roots of aox n (7)alxn-1+ (z)a2xn-2-...
=0, are non-unitary symmetric functions of the roots of a xn-a l xn 1 a2 x n-2 -...
It is thus possible to study simultaneously all the theories which depend upon operations of the group. Symbolic Representation of Symmetric Functions.-Denote the s 8 s elementar symmetric function a s by al a 2 a3 ...at pleasure; then, Y y si,, si,...
Denote by brackets () and  symmetric functions of the quantities p and a respectively.
Being subsequently put equal to a, a non-unitary symmetric function will be produced.
Symmetric Functions Several Systems Quantities.
1+Eaix+Esiy+ /al a2x 2 +Malt2xy -Z01023,2+ï¿½ï¿½ï¿½ The most general symmetric function to be considered is E 41 041 8424-3033..ï¿½ .conveniently written in the symbolic form (pigi p2g2 p3go...)ï¿½ Observe that the summation is in regard to the expressions obtained by permuting then suffixes I, 2, 3, ...n.
The weight of the function is bipartite and consists of the two numbers Ep and Eq; the symbolic expression of the symmetric function is a partition into biparts (multiparts) of the bipartite (multipartite) number Ep, Eq.
All symmetric functions are expressible in terms of the quantities ap g in a rational integral form; from this property they are termed elementary functions; further they are said to be single-unitary since each part of the partition denoting ap q involves but a single unit.
It will be ï¿½ shown later that every rational integral symmetric function is similarly expressible.
DaP4 References For Symmetric Functions.-Albert Girard, In- -vention nouvelle en l'algebre (Amsterdam, 1629); Thomas Waring, Meditationes Algebraicae (London, 1782); Lagrange, de l'acad.
1852; MacMahon, " Memoirs on a New Theory of Symmetric Functions," American 1 Phil.
1888-1890; " Memoir on Symmetric Functions of Roots of Systems of Equations," Phil.
Every symmetric function denoted by partitions, not involving the figure unity (say a non-unitary symmetric function), which remains unchanged by any increase of n, is also a seminvariant, and we may take if we please another fundamental system, viz.
It was noted that Stroh considers Method of Stroh.-In the section on " Symmetric Function," (alai +a 2 a 2 +...
Remark, too, that we are in association with non-unitary symmetric functions of two systems of quantities which will be denoted by partitions in brackets ()a, ()b respectively.
Twinning according to the second law can only be explained by reflection across the plane (roi), not by rotation about an axis; chalcopyrite affords an excellent example of this comparatively rare type of symmetric twinning.