tective is allied to that of the police, and wherever a police force exists there is some detective work to be done, though only in connection with a large police force are men regularly assigned to detective work. The police force of New York includes a body of men known as detective sergeants, who have charge of the work of looking for criminals or investigating such crimes as seem to call for their services. The United States government maintains a force, known as Secret Service men, whose principal duties consist in unearthing counterfeiters, and those who rob the mails or infringe the revenue laws. The British government has established in London a force of detectives known as Scotland Yard men. There are private detective establishments in all large cities, the best-known of these being the Pinkerton bureau, which has offices in several cities of the United States under the style of the Pinkerton National Detective Agency. This agency and similar bureaus make a business of supplying detectives, usually to get evidence in civil or criminal suits.

The private detective has fallen into some disrepute in the United States, owing to employment on divorce cases or other matters where there is a temptation to manufacture evidence instead of finding it. Some judges have refused to credit the testimony of such detectives unless corroborated. In many cities private detectives are obliged to take out a license before they are allowed to follow the calling. See POLICE; SECRET SERVICE.

DETENTION HOMES, State establishments provided by the juvenile court laws of the United States for the temporary care of children awaiting court decisions. They are usually located in rented private houses, some exclusively for delinquent children, but most, both for dependents and delinquents, the sexes separated, generally under the care of a husband and wife, or a matron assisted by male officers. In Syracuse, Columbus and Buffalo, rented, private houses contain both detention home and juvenile court. In Philadelphia, Milwaukee and Chicago special buildings accommodate under one roof waiting rooms, detention home and court rooms. See CHILDREN'S COURTS; JUVENILE OFFENDERS.

DETERMINANTS, an important class of algebraic functions which owe their origin to the attempt to formulate the solutions of gencral systems of simultaneous linear equations. Such a system of the second order is

a1x+b1y=k1, awx+bry=ks;

formed in accordance with a general principle, the first precise statement of which was based upon the recognition of the two classes of permutations, as will presently be explained.

2. It is shown in algebra (q.v.) that the number of permutations of n elements arranged in a series is n(n—1. . . 21n! Any two elements, whether adjacent or not, standing in their natural order in a permutation constitute a permanence; standing in an order the reverse of the natural, an inversion. Thus, in the permutation deacb the permanences are de, ac, ab; the inversions, da, dc, db, ea, ec, eb, cb.

The permutations of any set of elements are divided into two classes, viz.: the positive, in which the number of inversions is even, and the negative, in which the number is odd. When the elements are arranged in the natural order the number of inversions is zero, which is even.

3. Interchanging two adjacent elements, a and a, of a permutation changes its class. For, if a a is a permanence, a a is an inversion and vice versa; and the interchange either introduces or destroys an inversion. When the two elements interchanged are nonadjacent let the number of elements between them be q and represent these, in the aggregate, by Q. As in the preceding case the interchange has no effect upon the relation of a and a to the elements preceding or following aQa. The arrangement Qaa may now be obtained by interchanging a with each of the q elements of Q in turn, after which a may be moved to the first place by successive interchanges with the q+1 elements of Qa. Hence, the total number of interchanges of adjacent elements involved in the transition from the order aQa to the order a Qa is 2g+1, an odd number; from which follows the important theorem The interchange of any two elements of a permutation changes its class.

Of any complete set of permutations onehalf are positive and one-half negative. 4. Assume n' elements arranged in a square array thus: ai'ai" . . . a1(n) ar'ar" as(n)

"

In this array the position of any element is shown by its indices. For examples, as is in the third column and the fifth row. The diagonal through ai, a1⁄2",. an(n) is called the principal diagonal; that through an', an—1, a(n) the secondary diagonal; the position occupied by an the leading position.

The above array, enclosed by vertical bars as shown, is used to represent the determinant of its n2 elements. This function may now be defined.

Write down the product of the n elements on the principal diagonal, arranging them in the natural order, thus: aia'aa' ..an(n). This is the principal term of the determinant. Now permute the subscripts of the principal term in every possible way, leaving the superscripts undisturbed. To such of the n! resulting terms as involve the positive permutations of the subscripts give the plus sign; to those involving the negative permutations, the minus sign. The algebraic sum of all the terms thus obtained is the determinant represented by the given

array.

Each term of a determinant thus contains a single element from each column and each row of its array and is, therefore, a homogeneous function of its elements.

5. The expansion of the array of the second order may be written out at a glance. The process is less obvious, but still simple, for the array of the third order. It is as follows: Beneath the square array write the first and second rows as shown in the figure. Then form the six products, each of three elements, traversed by one of the six oblique lines, applying the signs as indicated. The aggregate of terms thus obtained is the required expansion, as may readily be verified.

The reader will now do well to note how the values of the systems of unknowns x, y, and x, y, 2, obtained at the outset, may be written in the notation of determinants.

No such direct methods as the above are available for the expansion of determinant arrays of higher orders, but these will be considered further on. See 13.

6. In writing determinants it is often convenient to use a double-subscript notation, the first subscript designating the row and the second the column to which the element belongs. Thus the element ass stands in the third row and the fifth column. When the elements are merely symbolic it is customary to write only the principal terms between the vertical bars. In this, which is called the umbral notation, the determinant of the nth order is

| al'ar" . . . an(n) | or | auɑn... ann |; which are often further abridged to | a(n) | and ain respectively.

Thus far the economy of the notation of determinants is scarcely apparent. Specific forms of higher order have, however, been purposely avoided. It is only necessary to write out the expansion of an array of the fourth order, which includes 41-24 terms each of the fourth degree, to understand the necessity of a general theory of such forms. Determinants of even the fifth and sixth orders would be, if written out in full, quite beyond manipulation; while the complete expansion of

| aran′′as" "a‚1vasva1va,va ̧VIIIa,Xα10×ɑ1X1ɑ12XII |, and such functions are not at all uncommon, would fill over a thousand closely printed volumes like the present! Yet, by means of the theory of determinants, such expressions are not only intelligible but manageable. The general properties of determinants will now be considered.

7. Any term of the development of a(n) may be written

±an'ar'aj" . . . ai(n). (a) Designate by u the number of inversions in the permutation hij...I and by v the number of interchanges of two elements necessary to bring the given term into the form

Obviously u and v are either both even or both odd; but the permutation pqr t is positive or negative, according as v is even or odd, and the term will, therefore, have the same sign whether it be determined by the permutation of the subscripts of (a) or by that of the superscripts of (b). It follows that the development of a determinant may be obtained by permuting the superscripts and writing the signs of the terms in accordance with these permutations, instead of using the subscripts as already explained. Passing from one of these methods of development to the other is equivalent to changing each column of the array into a row of the same rank and vice versa. Hence, a determinant is not altered by changing the rows into corresponding columns and the columns into corresponding rows. Any statement made with reference to the rows of every determinant must, therefore, be equally true with respect to the columns. Rows and columns are alike called lines.

8. If any two parallel lines of a determinant be interchanged the determinant will be changed only in sign. For, interchanging two lines is the same as interchanging, in each term of the expansion, the indices corresponding to these lines. This reverses the sign of each term and therefore that of the whole determinant.

The element a(8) may be transferred to the leading position by interchanging the kth row with the (1) preceding rows and the sth column with the (s-1) preceding columns. This being done, the resulting determinant must take the sign factor (-1) +8.

A determinant having two parallel lines identical is equal to zero; for the interchange of these identical lines reverses the sign without altering the value of the function.

9. A determinant having a line of elements each the sum of two or more quantities can be expressed as a sum of two or more determinants. Let

10. Multiplying each element of a given line of a determinant by a given factor multiplies the determinant by that factor; for each term of the expansion contains a single element from the given line. The common factor thus appears once and only once in each term of the expansion, and the determinant is, therefore, multiplied by that factor.

In the same way it may be shown that a determinant having a line of zeros is equal to zero. It also follows that if the elements of any

which is therefore the aggregate of the terms of | a1(n) |, (n−1)! in number, which contain the element a.

The determinant factor of order (n-1) by which the element a' is multiplied in (b) is called the cofactor of that element in a1(n) |. It may be obtained from the given determinant by deleting the first column and the first row.

The cofactor of any element ak(s) may be found in the same manner after transposing this element to the leading position. But this transposition multiplies the determinant by the sign factor (-1)+ Hence, to find the cofactor of ax(), delete its row and its column and give the resultant determinant the

(positive) sign when (<+s) is
(even).

\negative/

The cofactor thus obtained is represented by AK(8), the sign factor (-1)x+8 being intrinsic. For example, the cofactors of the elements of the second row of | aa"a"" | are ar'a,""

A=- | aa", |, A = aa"

aa

A""a

13. The aggregates of terms containing the elements ak, ak", . a(n) of the determinant

...

| a(n) | are, respectively,

an' A'‚ak"AK", . . . a (n) A ̧(n).

Each of these n aggregates includes (n-1)! terms of a(n) |, no one of which appears in any of the others. In all of them, then, there are n(n-1)! or n!, different terms of the determinant, which is the whole number. Hence

|a, (n)=ax'A' +ax"AK" + ... + a(n) A(n) (1) Similarly,

|a1(n)|=a1(8) A1(8) + as(8) A2(8) + +an(8) An(8) (2)

Any determinant may, by means of either (1) or (2), be resolved into determinants of an order one lower and thus, since AK',... A(n) or A(), ... An() are themselves determinants, it may ultimately be expressed in terms of deter

If two rows, the hth and xth, and two columns, the pth and sth, are deleted the result is written 4h,k(P,8), and is called a minor of the second order. Minors of lower orders may be obtained in a similar manner and expressed by a similar notation.

Any mth minor of a given determinant and the determinant of the m3 elements at the intersection of the rows and columns deleted in forming it are called, with respect to each other, complementary minors. The determinant may be expressed in terms of products of pairs of complementary minors, a method of expansion due to Laplace. Formulæ (1) and (2) are special cases of the method. Its general statement is somewhat complicated.

16. The principles thus far developed will now be applied to the solution of systems of simultaneous linear equations; the process which, as stated at the outset, led to the discovery and investigation of determinants. Assume the system of three equations

aix+biy+ciz=Ki. (i=1, 2, 3.)

In the determinant | Kibaca | let the elements K1, K2, K1 be replaced by the equal quantities appearing in the first members of the given equations. The two determinants now in hand are equal to each other; thus

does not vanish the equations are satisfied only by the values ry 0. In general: The condition that n homogeneous linear equations between unknowns form a consistent system, for other than zero values of the unknown, is that the determinant of the coefficients be zero.

The relation expressed by (7) may also be generalized thus: In any determinant which equals zero, the cofactors of the elements of any line are proportional to the cofactors of the corresponding elements of any parallel line. The determinants (5) and (6) are called resultants or eliminants, each being the result of climinating the unknowns from the system of equations from which it is derived.

19. Again let there be r homogeneous linear equations involving n unknowns, being greater than n, thus:

ai'x' +ai"'x" + +ai(n)x(n): = 0.

...

(i=1, 2, ni,. r).

...

The consistency of these equations requires that every determinant of the nth order, formed by selecting n rows from the array whose elements are the coefficients written in order, shall be If these conditions are fulfilled the fact is expressed by writing

zero.

in which case each unknown becomes

0 may have any value whatever. But the given equations may be written

y + bi 2 + ci=0;

and

(i = 1, 2, 3) any two of which will determine the ratios 9 .

If these three equations form a con

If the given equations be inconsistent this determinant does not vanish.

This process, due to Sylvester, may readily be generalized. It is known as the dialytic method of elimination.

21. The product of two determinants may be most readily obtained as an eliminant. To this end let

a11x1+a12x2=0, A21x1 + A22x2 = 0 be linearly transformed by substituting x1=b1w1+b21w2, X2=b12w1 + b1ws; the result being

(a)

(b)

The vanishing of either of the determinants (e) or (f), therefore, causes (d) to vanish; from which it follows that the determinants (e) and (f) are factors of (d). The only other factor is numerical and may readily be shown, by comparing coefficients, to be unity. Hence, a11a12 b11b12 +a12b12 a11b21 + a12b22 baba anbu

a21b11 +a22b12 ɑ21b21 + ɑ22b22 | The same method may be applied to the formation of the product of any two determinants of the same order. The operation may be described as follows:

To form the product pin of two determi nants ain and bin, first connect by plus signs the elements of the rows of both an and bin. Then place the first row of lain upon each row of bin in turn and let each two elements as they touch become products. This is the first row of Pin. Perform the same operation upon bin with the second row of lain to obtain the second row of pin, etc. Any element of this product is

That is the reciprocal of a determinant of the nth order is equal to its (n-1)th power.

Reciprocal determinants are a special case of compound determinants, whose elements are minors variously derived from one or more given arrays.

24. The application of determinants to the study of geometry and pure analysis has led to the recognition and investigation of numerous special forms, some of which will be defined and their most characteristic properties stated without demonstration.

Among the more important are the forms known as symmetrical determinants. In these any two elements symmetrically placed with respect to the principal diagonal, known as conjugate elements, have the same absolute value. If a(s) = ag(k) the determinant is described as simply symmetrical. If a¿(®)=—a ̧(K), a condition which cannot apply to the elements of the principal diagonal, unless these be zero, it is said to be gauche or skew. When the diagonal elements are zeros and a(8)=A8(K) the determinant is skew-symmetric. It is very easy to show that a skew-symmetric determinant of odd order is equal to zero. When of even order, however, it is equal to the square of a certain rational function of its elements known as the Pfaffian. These functions are expressed by triangular arrays; thus

P8=abs +a2b82 The product may also be formed by columns instead of by rows as above.

22. The operation just described may be applied to form what is conventionally called the product of two rectangular arrays of the same dimensions. Let these be

= (ca — nl + bm)2= |anm | 2. bl

C

The properties of Pfaffians are strikingly analogous to those of determinants.

25. Determinants all of whose elements are zeros except those of the prinicipal diagonal and the adjacent minor diagonals above and below, and in which each element of one of these minor diagonals is —1, are called continuants. They were so named by Muir because of their connection with the theory of continued fractions. If qa be the number of terms in the expansion of the continuant of order i, it may be shown that qnqn-1+9n—2; a difference equation, the solution of which is (1+V5)”+1—(1—V 5)n +1 2n+1 V 5

Qn=

1

26. A determinant in which the elements of the first row are functions of a given variable, the corresponding elements of the second row the same function of another variable, etc., is I called an alternant. If the functions used as elements are powers of the variables it is described as a simple alternant. Such a determinant is divisible by the difference product of its variables, the quotient being a symmetrical function of these variables. Thus

1 x x=(y-z) (z-x) (xy). (x + y + z).
1 yys