And there is no originality in it; only endless **permutations** and combinations of doctrines already known and accepted.

**Permutations** and Combinations.

Every unit of the rth species which does not vanish is the product of r different units of the first species; two such units are independent unless they are **permutations** of the same set of primary units e i, in which case they are equal or opposite according to the usual rule employed in determinants.

In particular, the number of **permutations** of the whole stock is n!.

Y ...a n v, the summation being for all **permutations** of the n numbers, is called the determinant of the n 2 quantities.

Now a11A11= alla22a33...ann, wherein all is not to be changed, but the second suffixes in the product a 22 a 33 ...a nn assume all **permutations**, the number of transpositions necessary determining the sign to be affixed to the member.

(v.) **Permutations** and Combinations may be regarded as arithmetical recreations; they become important algebraically in reference to the binomial theroem (ï¿½ï¿½ 41, 44)ï¿½ (vi.) Surds and Approximate Logarithms. - From the arithmetical point of view, surds present a greater difficulty than negative quantities and fractional numbers.

The number (n P r) of **permutations** of r individuals out of a stock of n, all being distinguishable, is n`r).

The number of distinguishable **permutations** of the whole stock is n!= (a?b!cI (ii.) A combination is a group of individuals without regard to arrangement.

The intervention of the Curia, which hitherto had been hostile to Casimir because of his steady and patriotic resistance to papal aggression, was due to the **permutations** of European politics.

Under the general heading "Algebra and Theory of Numbers" occur the subheadings "Elements of Algebra," with the topics rational polynomials, **permutations**, &c., partitions, probabilities; "Linear Substitutions," with the topics determinants, &c., linear substitutions, general theory of quantics; "Theory of Algebraic Equations," with the topics existence of roots, separation of and approximation to, theory of Galois, &c. "Theory of Numbers," with the topics congruences, quadratic residues, prime numbers, particular irrational and transcendental numbers.

**Permutations** of the n numbers I, 2, 3, ...