A separation is the symbolic representation of a product of monomial symmetric functions.
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.
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!
P operators D upon a monomial symmetric function is clear.
D p4sp4 +Dp4(pg)+1; d P4 causes every other signle part function to vanish, and must cause any monomial function to vanish which does not comprise, one of the partitions of the biweight pq amongst its parts.
Since dp4+(-)P+T1(p +q qi 1)!dd4, the solutions of the partial differential equation d P4 =o are the single bipart forms, omitting s P4, and we have seen that the solutions of p4 = o are those monomial functions in which the part pq is absent.
- (i.) An expression such as a.2.a.a.b.c.3.a.a.c, denoting that a series of multiplications is to be performed, is called a monomial; the numbers (arithmetical or algebraical) which are multiplied together being its factors.
(ii.) By means of the commutative law we can collect like terms of a monomial, numbers being regarded as like terms. Thus the above expression is equal to 6a 5 bc 2, which is, of course, equal to other expressions, such as 6ba 5 c 2.
In order that a monomial containing a m as a factor may be divisible by a monomial containing a p as a factor, it is necessary that p should be not greater than m.
(viii.) The quadratic equation is the equation of two expressions, monomial or multinomial, none of the terms involving any power of x except x and x 2 .