The operation may be effected by giving fractional values to n in equation (1), Art. 589.

595. 1. Given √52.2361, √6 = 2.4495, √7 = 2.6458, √8 = 2.8284, ...; find √6.3.

In this case the successive orders of differences are:

Since the required term is distant 1.3 intervals from √5, we have n = 2.3.

Substituting in (1), Art. 589, we have, approximately,

2. Given log 22 = 1.3424, log 23= 1.3617, log 24 = 1.3802, log 25 1.3979, ...; find log 24.5.

=

3. Given 704.12129, 714.14082, 724.16017, ...; find 70.12.

4. The reciprocal of 22 is .04545; of 23, .04348; of 24, .04167; etc. What is the reciprocal of 22.8?

=

5. Given log 109 2.03743, log 110 2.04139, log 111 110=2.04139, = 2.04532, .; find log 110.7.

...

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6. Given √37 6.08276, √38 = 6.16441, √39 = 6.24500, ...; find √37.48.

==

=

7. Given log 11 = 1.04139, log 12 1.07918, log 13 = 1.11394, log 14 1.14613, ...; find log 13.28.

which is understood as signifying the product of the upper left-hand and lower right-hand quantities, minus the product of the lower left-hand and upper right-hand.

The expression (1) is called a Determinant of the Second Order.

597. The numerators of the fractions in the preceding article can also be expressed as determinants; thus,

The denominator of (1) may be written in the form

which is understood as signifying the sum of the prod ucts of the quantities connected by lines parallel to a line joining the upper left-hand corner to the lower right-hand, in the following diagram, minus the sum of the products of the quantities connected by lines parallel to a line joining the lower left-hand corner to the upper right-hand.

The expression (2) is called a Determinant of the Third Order.

599. The numerator of the value of x can also be expressed as a determinant, as follows:

as may be verified by expanding it by the rule of Art. 598

601. General Definition of a Determinant.

If, in any permutation of the numbers 1, 2, 3, ..., n, a greater number precedes a less, 'there is said to be an inversion.

Thus, in the case of five numbers, the permutation 4, 3, 1, 5, 2, has six inversions; 4 before 1, 3 before 1, 4 before 2, 3 before 2, 5 before 2, and 4 before 3.

Consider, now, the n2 quantities

Note 1. The notation in regard to suffixes, in the above, is that the first suffix denotes the horizontal row, and the second the vertical column, in which the quantity is situated.

Thus, a,r is the quantity in the kth row and rth column.

Let all possible products of the quantities taken n at a time be formed, subject to the restriction that each product shall contain one and only one quantity from each row, and one and only one from each column, and write them so that the second suffixes shall occur in the order 1, 2, n.

Note 2. This is equivalent to writing all the permutations of the order 1, 2, n in the first suffixes.

Give to each product the sign + or according as the number of inversions in the first suffixes is even or odd. The expression (1) is called a Determinant of the nth Order.

602. The expanded form may also be obtained by writing the first suffixes in the order 1, 2, ..., n, and giving to each product the sign + or according as the number of inversions in the second suffixes is even or odd.

For let the absolute value of any product, obtained as in Art. 601, be

ap, 1 aq, 2... Ar, nj

where p, q,..., r is a permutation of 1, 2, Writing the first suffixes in the order 1, 2,

(1)

..., n, we have

(2)

where s, t, ..., v is a permutation of 1, 2, ..., n.

It is evident that there are just as many inversions in the first suffixes of (1) as in the second suffixes of (2); and hence the products (1) and (2) will have the same sign.

603. The quantities a,1, α1, 2, etc., are called the constituents of the determinant, and the products a1, a2, 2... an, n etc., occurring in the expanded form, are called its elements.

The constituents lying in the diagonal joining the upper left-hand corner to the lower right-hand, are said to be in the principal diagonal; the element whose factors are the constituents in the principal diagonal is always positive.

Note. By Art. 543, the number of elements in the expanded form of a determinant of the nth order is │n.