For the subjects of this general heading see the articles ALGEBRA; ALGEBRAIC FORMS; ARITHMETIC; **COMBINATORIAL** ANALYSIS; DETERMINANTS; EQUATION; FRACTION, CONTINUED; INTERPOLATION; LOGARITHMS; MAGIC SQUARE; PROBABILITY.

X3 (ll'1T2...) and it is seen intuitively that the number 0 remains unaltered when the first two of these partitions are interchanged (see **Combinatorial** Analysis).

It is shown in the article on **Combinatorial** Analysis that (w; 0,n) is the coefficient of a e z w in the ascending expansion of the fraction 1-a.

Algebraic Forms; Binomial; **Combinatorial** Analysis; Determin Ants; Equation; Continued Fraction; Function; Theory of groups; Logarithm; Number; Probability; Series.

- The discussion, in ï¿½ of the number of terms of a particular kind in a particular product, forms part of the theory of **combinatorial** analysis, which deals with the grouping and arrangement of individuals taken from a defined stock.

(iv.) Other developments of the theory deal with distributions, partitions, &c. (see **Combinatorial** Analysis).

The multi- (or poly-) nomial theorem has for its object the expansion of any power of a multinomial and was discussed in 1697 by Abraham Demoivre (see **Combinatorial** Analysis).

See further **COMBINATORIAL** ANALYSIS.

Pfaff's researches bore chiefly on the theory of series, to which he applied the methods of the so-called **combinatorial** school of German mathematicians, and on the solution of differential equations.

The mathematical theory of probability and the allied theory of the **combinatorial** analysis were in effect created by the correspondence between Pascal and Fermat, concerning certain questions as to the division of stakes in games of chance, which had been propounded to the former by the gaming philosopher De Mere.