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 .