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DESPOTO DAGH - DETECTIVE

toward the end of the 12th century, is said to quarters of the rebellion. Pop. est. about have first introduced this title, and to have 15,000. made it the first in rank after that of emperor. Destouches, Philippe Néricault, fé-lēp nå. Thus there was a despot of the Morea, of Şer- re-kö, French dramatist: b. Tours, France, via, etc. At present, despot means an absolute 22 Aug. 1680; d. near Melun, France, 4 July ruler, as the emperor of Russia ; but, in a nar

1754. His comedy, (The Boaster) (1732), is a rower sense, it conveys the idea of tyranny, as in masterpiece in matter, in elaboration, and in fact the possession of absolute power and the character delineation; Lessing classes that work, abuse of it are two things bordering very closely with its companion piece, The Spendthrift on each other.

(1736), as models of the finer high comedy.) Despoto Dagh, děs po to däh, a mountain Hardly inferior to these is (The Married Philoschain of European Turkey, extending from 30 opher? (1727), largely based on the author's miles to the east of the Balkans to the bank own life. of the Maritza.

Destroying Angels. See DANITES. Dessaix, Joseph Marie, zho-zěf mä-rē dā. Destutt de Tracy, Antoine Louis Claude, sā, French general: b. Thonon, Savoy, 24 än-twän loo ė klod dā-stüt de trä-sē, COUNT, Sept. 1764; d. 26 Oct. 1834: He served at the French philosophical and metaphysical writer : siege of Toulon, and in Italy under Bonaparte; b. Paris 20 July 1754; d. there 10 March 1836. was elected in 1798 to the council of 500, where Though in repeated peril during the French he opposed the coup d'état of the 18th Brumaire. Revolution, he survived to write Elements of He was made a brigadier-general by Bonaparte Ideology? (1817), a development of Condillac's in 1803, and, in the campaign of 1809 against philosophy, and in part an exposition of what Austria, a general of division, receiving

from then

passed for economics. His Delineations of the emperor the surname of L'intrepide, and the the Politics of the World's Nations) (1820), and title of count of the empire.

prior works, received considerable notice in the Dessalines, Jean Jacques, zhon zhäk dá-sa- United States through Jefferson, who translated lên, emperor of Haiti: b. 1760; d. 14 Oct. the Commentaire sur l’Esprit des Lois) (1806) 1806. He took the name of the person in into English and had it published in Philadelphia whose service he remained until 1790; after that (1811). time he fought under Biasson, and, still later, Desulto'res (from desilio, "Ivault”), the joined Toussaint L'Ouverture. In his struggle Latin name for vaulters or leapers, who jumped against Gen. Rigaud he signalized himself as

from one horse to another. The Scythian, Inmuch by his cruelty as his bravery. In 1802 he dian, and Numidian cavalry were very expert surrendered to Gen. Leclerc. But when an epic desultores, and each man carried at least two demic of yellow fever fell upon the French horses to the field. \Vhen one was weary he army and almost annihilated it, he attacked jumped with great agility upon another, which Rochambeau with an army of 30,000 blacks, thus he led by his hand. The Greeks and Romans obliging the French commander to surrender introduced the same practice in their games, to the English, and to leave the island (1803). races, and funeral solemnities, but never, as far In 1804, when governor-general of Haiti, he

as we know, in war. Homer describes a vaulter issued an order for the general slaughter of of this sort who performed his feats on four the white inhabitants. In October of the same

horses at once (Iliad, xv. 679); and (xxiii. year he was proclaimed emperor, and made an 29) describes a kind of Numidian cavalry in unsuccessful attempt to take the city of Santo Hasdrubal's army in Spain, in which the sol Domingo in March 1805. Incurring the enmity diers had two horses each, and in the heat or of his own followers, he was killed in an am

an engagement frequently leaped, fully armed, buscade near Port au Prince.

from one to another. Ælian gives a similar Dessau, děs'sow, Germany, capital of the account of a tribe dwelling not far from the duchy of Anhalt, in a valley on the Mulde, on Danube, who, on this acco int, were called Amthe railroad between Berlin, Köthen, and Leipsic. phippi. The principal building is the ducal palace, built Detachment, a body of troops or part of in 1748, containing both a picture-gallery and a a fleet selected from the main body for some library, in which are numerous MSS. of Luther. special service. The manufactures consist of woolen and linen

Detaille, Jean Baptiste Edouard, zhõn bäpcloth, hats, leather, tobacco, musical and other tēst ěd-oo-ärd, de-tä-yė, French painter: b. instruments. The ground around Dessau, orig- Paris 5 Oct. 1848. He is distinguished for his inally a sandy waste, has been completely reclaimed, and is now covered with beautiful gar- of his best pictures, The Passing Regiment,'

treatment of battles and military subjects. One dens. Pop. 42,375.

is in the Corcoran Art Gallery in Washington, Dessicants, in medicine, substances that D. C. check secretions from mucous membranes or

Detective, one who searches for criminals cause cicatrization. See ASTRINGENTS.

or ferrets out crime. The work of the detecDesterro, dās-tãr'ro, also called officially tive is allied to that of the police, and wherever Florianopolis, Brazil, the capital of the state a police force exists there is some detective work of Santa Catharina. It is situated on a long to be done, though only in connection with a and narrow island near the mainland, and its large police force are men regularly assigned foreign trade passes through the port of São to detective work. The police force of New Francisco, which is one of the best on the York includes a body of men known as detective coast of southern Brazil. During the revolt sergeants, who have charge of the work of of the navy in 1893, the most critical period looking for criminals or investigating such through which the new Brazilian institutions crimes as seem to call for their services

. The have passed, Desterro and its port were head- United States government maintains a foree,

DETERMINANTS

known as Secret Service men, whose principal he is almost certain to elude the pursuer. The duties consist in unearthing counterfeiters, and detective who follows one must not stop whenthose who rob the mails or infringe the revenue ever his man stops, but go right on and appear laws. The British government has established to pay no attention to him. He must know in London a force of detectives known as Scot- enough to jump on a car to get ahead of his man, land Yard men.

and to be out of view; to dodge through a shortThere are private detective establishments cut if there be one where he cannot lose his in all large cities, the best-known of these being man, or to do any one of a dozen things promptly the Pinkerton bureau, which has offices in sev- to serve his purpose. It is dangerous to take eral cities of the United States under the style his eyes off the crook, and equally dangerous to of the Pinkerton National Detective Agency. let the crook have a good look at his shadower, This agency and similar bureaus make a business so that if the following has to be kept up for

any of supplying detectives to any one who will pay considerable distance it is a very trying piece for the work, usually to get évidence in civil or of work. criminal suits.

The private detective has fallen into some The detective achieves success by studying the disrepute in the United States, owing to employways of lawbreakers, and becoming acquainted ment on divorce cases or other matters where with the haunts of the men he seeks. For in- there is a temptation to manufacture evidence stance, William Pinkerton, of Chicago, made an instead of finding it. Some judges have refused exhaustive study of the class of tramps who rob to credit the testimony of such detectives unless country stores and post-offices, and blow off the corroborated. In many cities private detectives doors of safes. Their own name for their class are obliged to take out a license before they are is "yeggmen, and the only way to know them allowed to follow the calling. thoroughly is to become one of them for a time, The Secret Service men whose duty it is to to live as they do, and thus secure their con- protect the mails have a simple method of locatfidence, and gain familiarity with their system ing thieves. Whenever a letter is lost, and a of exchanging information. These yeggmen had tracer sent after it, and the man found who last a habit of registering on the tank houses of had knowledge of the letter, a pin is stuck in a railway stations, where they would write the map at the city where the letter was last seen. names by which they were known, with the date, As more lost letters are searched for, the pins and the direction they were going. Doubtless in this map begin to show central points where this practice has changed for some other, since letters disappear with regularity. This shows the detectives became familiar with it, and used about where there is regular thieving, and a their knowledge to lodge various bad characters detective has his field of work clearly pointed behind the bars.

out to him, and by watching the men who handle Those detectives who achieve the best results the mail in that district he is usually sure of make a specialty of one line of work, with which locating his man before very long. See also they become perfectly familiar, In a city like Police; Secret SERVICE. New York there is on the detective force at least one man familiar with the ways of each of functions which' owe their origin to the attempt

Determinants, an important class of algebraic the principal class of crooks. If a pickpocket is wanted, the man familiar with that work is

to formulate the solutions of general systems of

Such a system supposed to have a pretty good idea what pick

simultaneous linear equations. pockets are in town, and where they can be of the second order is located; if he be an extra good detective, he

a,x +by=k, ax + b2y =Kz; will also be able to form a judgment from the nature of a steal, who are most likely to have from which been the thieves. It is this sort of knowledge that makes possible the quick detection of crimi

к,b, к,b, 2,K2 - A,K,

y = nals.

a, b, -a,bi' a, b, - a,b, The nature of detective work is such that the solution of the system of the third order, very little authoritative matter has been printed about it. One of the first requisites of a de

aix + bay +C;2 = kj, (i=1, 2, 3), tective is the power of keeping silence about in like manner gives the business or the cases on which he works. There are no printed reports and statistics of к,b,c, +к,b,c, +к,b,c, к,b,c, к,b,c, - к,,c, detective work that could be used in an encyclo- a,bzcz + a,b,c, +a,b,c, –a,b,c, - a,bc, - a,b,cz'. pædia. Most of the literature is of the nature of detective stories, told to amuse, and con- with expressions of similar form for y and .. cealing instead of making clear real occurrences The functions appearing in the numerators or facts. The detective at work is really a very and denominators of the expressions for the different man from what he is pictured in the unknowns in the above, and in similar systems novel. He has usually had police experience, of equations, determinants. They are and is big and strong: These things, with a formed in accordance with a general principle, knowledge of the class of men he is to seek, are the first precise statement of which was based his stock in trade. One of the most common upon the recognition of the two classes of perand difficult jobs that falls to his lot is the simple mutations, as will presently be explained. following of a man, whom he has located, but 2. It is shown in algebra (q.v.) that the numwhom he does not wish to arrest, until he has ber of permutations of n elements arranged traced him to his living place or some haunt non-cyclically is n(n-1). ..2.1 =n!. Any two where he can also expect to locate some com- elements, whether adjacent or not, standing in panion or accomplice. To follow a man in an their natural order in a permutation constitute ordinary way is to invite him to escape, for if a permanence; standing in an order the reverse a crook’in a city street suspects he is followed, of the natural, an inversion. Thus, in the per

are

DETERMINANTS

See 13

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a'a,

a,(n)

mutation deach the permanences are de, ac, ab; Each term of a determinant thus contains a the inversions, da, de, db, ea, ec, eb, cb.

single element from each column and each row The permutations of any set of elements are of its array and is, therefore, a homogeneous divided into two classes, viz.: the positive, in function of its elements. which the number of inversions is even, and 5. The expansion of the array of the second the negative, in which the number is odd. order may be written out at a glance. The When the elements are arranged in the natural process is less obvious, but still simple, for the order the number of inversions is zero, which array of the third order. It is as follows, the is even.

columns being, in this instance, ranked alpha3. Interchanging two adjacent elements, a and betically instead of by indices: Beneath the a, of a permutation changes its class. For, if square array write the first and second rows as aa is a permanence, aa is an inversion and vice shown in the figure. Then form the six products, versa; and the interchange either introduces or each of three elements, traversed by one of the destroys an inversion. When the two elements six oblique lines, applying the signs as indicated. interchanged are non-adjacent let the number The aggregate of terms thus obtained is the of elements between them be 9 and represent required expansion, as may readily be verified. these, in the aggragate, by Q. As in the pre- The reader will now do well to note how the ceding case the interchange has no effect upon values of the systems of unknowns x, y, and x, the relation of a and a to the elements preceding y, z, obtained at the outset, may be written in or following aQa. The arrangement Qaa may the notation of determinants. now be obtained by interchanging a with each of No such direct methods as the above are the q elements of Q in turn, after which a may available for the expansion of determinant be moved to the first place by successive inter- arrays of higher orders, but these will be conchanges with the q+1 elements of Qa. Hence, sidered further on. the total number of interchanges of adjacent 6. In writing determinants it is often conelements involved in the transition from the venient to use a double-subscript notation, the order aQa to the order aQa is 29+1, an odd first subscript designating the row and the number; from which follows the important second the column to which the element belongs. theorem: The interchange of any two elements Thus the element as stands in the third row of a permutation changes its class.

and the fifth column. When the elements are Of any complete set of permutations one half merely symbolic it is customary to write only are positive and one half negative.

the principal term between the vertical bars 4. Assume na elements arranged in a square In this, which is called the umbral notation, the array thus:

determinant of the nth order is a,'a,"

19,'a," ...anch) or 144,072 ... Onni:

' anin|

which are often further abridged to la, (n) | and an'an" an(n)

Ja, respectively.

Thus far the economy of the notation of In this array the position of any element is determinants is scarcely apparent. Specific shown by its indices. For example, a,'"' is in forms of higher order have, however, been the third column and the fifth row. The diagonal purposely avoided. It is only necessary to through a,', a,"', :.. anin) is called the principal write out the expansion of an array of the diagonal; that through an', an-1", ... a, (n) the fourth order, which includes 41=24 terms each secondary diagonal; the position occupied by a,' of the fourth degree, to understand the necesthe leading position.

sity of a general theory of such forms. DeterThe above array, inclosed by vertical bars as minants of even the fifth and sixth orders would shown, is used to represent the determinant of be, if written out in full, quite beyond manipuits na elements. This function may now be lation; while the complete expansion of defined.

Write down the product of the n elements on Ja,'a,','',1va,va,vta, vila, villa,1x0,2*0,1x10,2x1, the principal diagonal, arranging them in the natural order, thus: a,'a,"az" ... anin). This and such functions are not at all uncommon, is the principal term of the determinant. Now would fill over a thousand closely printed volpermute the subscripts of the principal term in

umes like the present! Yet, by means of the every possible way, leaving the superscripts un

theory of determinants, such expressions are not disturbed. To such of the n! resulting terms only intelligible but manageable. The general as involve the positive permutations of the sub- properties of determinants will now be con

sidered. scripts give the plus sign; to those involving the negative permutations, the minus sign.

7. Any term of the development of a, (n) | may

be written The algebraic sum of all the terms thus obtained is the determinant represented by the

Fan'a:"a;'" ...az(n).

(a) given array.

Applying the process to the determinant Designate by u the number of inversions in the array of the second order, there results

permutation hij ...l and by v the number of

interchanges of two elements necessary to bring a,'a," =a'a," -a, a,";

the given term into the form
£a,Ma,(!)az(") ...0,(!).

(6) while that of the third order gives

Obviously u and v are either both even or both Ea, ardi

odd; but the permutation por ...t is positive a.

or negative, according as v is even or odd, and the term will, therefore, have the same sign

XII

,'

az'a."

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DETERMINANTS

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whether it be determined by the permutation ing elements of any parallel line the deterof the subscripts of (a) or by that of the super- minant vanishes. scripts of (b). It follows that the development 11. If each element of a line of a determinant of a determinant may be obtained by permuting be multiplied by a given factor and the product the superscripts and writing the signs of the added to the corresponding element of any terms in accordance with these permutations, parallel line the value of the determinant will instead of using the subscripts as already ex- not be changed. This follows directly from plained. Passing from one of these methods 9 and 10. Thus of development to the other is equivalent to changing each column of the array into a row

а.а,,..... а, 14...n

0,,2,,(0,8 +ma) aan of the same rank and vice versa. Hence, a determinant is not altered by changing the rows an, anang... Ann any anglang + man).i. Onn into corresponding columns and the columns

12. The terms of la,(n) | which contain the into corresponding rows. Any statement made element a,' are those found in the expansion of with reference to the rows of a determinant must, therefore, be equally true with respect

(a) to the columns. Rows and columns are alike

a, a,''a,'' called lines.

8. If any two parallel lines of a determinant be interchanged the determinant will be changed For if, in forming any term, another element than only in sign. For, interchanging two lines is the same as interchanging, in each term of a' be taken from the first column an element the expansion, the indices corresponding to these zero must be taken from the first row, and the lines. This reverses the sign of each term and térm vanishes. It may readily be shown that therefore that of the whole determinant.

the determinant (a) is equal to The element axls) may be transferred to the

a,"a,'" ...a,(n)

(6) leading position by interchanging the kth row

a' with the (x-1) preceding rows and the sth

nama

an's column with the (s-1) preceding columns, This being done, the resulting determinant must which is therefore the aggregate of the terms of take the sign factor (-1)**s.

la,(a), (n-1)! in number, which contain the A determinant having two parallel lines iden- element a,'. tical is equal to zero; for the interchange of

The determinant factor of order (n − 1) by these identical lines reverses the sign without which the element a, is multiplied in (6) is altering the value of the function.

called the co-factor of that element in Ja, (-) 1. 9. A determinant having a line of elements It may be obtained from the given determinant each the sum of two or more quantities can be by deleting the first column and the first row. expressed as a sum of two or more determinants. The co-factor of any element aq's) may be. Let

found in the same manner after transposing

this element to the leading position. But this A=14,(6, +b;'-6" )

transposition multiplies the determinant by the 0,(6, +, -6,

sign factor (1)**s. Hence, to find the coaj(ba+b3'. -63"

)C3

factor of ax(s), delete its row and its column and

give the resultant determinant the be such a determinant. Then, writing

(positiyee) sign when (a+s) is
sign when («+s) is (even).

)
B, = b; +b;'-b;" F ....,

The co-factor thus obtained is represented any term of the development of 4 is of the form by Axls), the sign factor (-1)+s being

intrin. #apBqcp.. • #apbace

For example, the co-factors of the #apbacr... Fapbe" cr....

elements of the second row of la,'a,"a,"") are 'a:''

'

la/'a," subscripts p, q,, of apBqCr... Permuting simultaneously the same subscripts in the 13. The aggregates of terms containing the second member and giving to each term thus elements ax', ax ax(n) of the determinant obtained its appropriate sign, there results ja,(n) | are, respectively, 14,B,C... Islam , = ,bcg...1

ax'Ar', ax"A", ... Qx(n.Amin!
+14,6 c...1-1a,b,"c,......,
baca

Each of these n aggregates includes (n − 1)! which proves the theorem.

terms of la, (n'l. no one of which appears in any 10. Multiplying each element of a given line of the others. In all of them, then, there are of a determinant by a given factor multiplies nin – 1)! or nl, different terms of the deterthe determinant by that factor; for each minant, which is the whole number. Hence term of the expansion contains a single element from the given line. The common factor thus

la'n' |- a ' Ax' + ax"A" +...+axin' A*) (1) appears once and only once in each term of the

Similarly, expansion, and the determinant is, therefore, multiplied by that factor.

|a,(*)]=a,(s) A, (s) +a,) A,'' +...+QAS) AS) (2) In the same way it may be shown that a determinant having a line of zeros is equal to Any determinant may, by means of either (1) zero. It also follows that if the elements of or (2), be resolved into determinants of an order any line have a common ratio to the correspond. one lower and thus, since An's....Aglas or

VOL. 6 - 44

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sic.

The terms of 4 are obtained by permuting the A'= -1;5), A," =

-13:, A," =64;7), A," =

Ja,'a

K

DETERMINANTS

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A,, ... Anls) are themselves determinants, it 14, Kzcal and fa,b,x31, respectively, the values of may ultimately be expressed in terms of deter- y and 2 are found to be minants of the third or second order, which may readily be expanded (see 5.).

|a, K,C31 Ja, b, ks 14. If the hth and throws of la,(n) | are

1a, baca la identical the elements ax'n ax",

arin) in

It will be noted that the values of the unformula (1) may be replaced by an', an", .. ann), knowns have for a common denominator the respectively. But in this case the value of the determinant of the coefficients of the given determinant is zero. Hence, h and « being equation; while the numerator is, in each case, different indices,

obtained from the denominator by replacing the ah'An' tan" A" +... takin) Ax(n) = (3) column of coefficients of the unknown in ques

tion by the column of absolute terms. The Likewise p and s being different,

method is applicable to linear systems of any

order. a,'PA,(s) + a,(P) A,(s) +... tanp) An(s) - 0. (4)

17. When the number of given equations is 15. The determinant of order (n − 1) obtained greater than the number of unknowns their conby deleting the ath row and the sth column of sistency obviously depends upon some definite A a, (n) | is called the minor of the determi- relation among the known elements. Let nant with respect to the element axls), and is

ajx + biy=Kj (i=1, 2, 3) written Acx's. Obviously, by what precedes,

be such a redundant system. A.(s) (-1)*+s. AC'S.

Solving the first two of these equations If two rows, the hth and kth, and two columns, gives (see 1) the pth and sth, are deleted the result is written

κό,

a, ki dih. xlpos, and is called a minor of the second

|а,к, | order. Minors of lower orders may be ob

y = tained in a similar manner and expressed by a

a,b, a, b,

laba similar notation.

a, b, Any mth minor of a given determinant and If, now, the three equations are consistent, the determinant of the mo elements at the inter- these values must also satisfy section of the rows and columns deleted in

a+bay=s; forming it are called, with respect to each other, complementary minors, The determi- whence, substituting the above values and clear. nant may be expressed in terms of products of ing of fractions, pairs of complementary minors, a method of expansion due to Laplace. Formulæ (1) and

az xbq+b3a, ky = K a,b, (2) are special cases of the method. Its general

к,b,

azba statement is somewhat complicated.

or, by 8, and (1) 16. The principles thus far developed will now be applied to the solution of systems of

- bala, ki 1 + xyla, b, simultaneous linear equations; the process

azba which, as stated at the outset, led to the dis

a, b,

(5) covery and investigation of determinants. Assume the system of three equations

(a, b, K2

lazbzK3! aix+biy +cis + ki. (i = 1, 2, 3.)

The above process being generalized, it appears In the determinant \«,b,cal, let the elements ky, that the condition that n linear equations bekg, is be replaced by the equal quantities appear tween (n-1) unknowns constitute a consistent ing in the first members of the given equations. system is that the determinant of the coeffiThe two determinants now in hand are equal cients and absolute terms be zero. to each other; thus

18. Consider now the homogeneous lineara

equations
19,x + b, y+6,5 b,c, ,b,g
09x +boy+b,c, к,b,c,

dix+biy + Cys=0. (i-1, 2, 3.)
Agx + bay + cg: bzia! K3O3C:

Solving these equations as in 16 gives But the first member of this equation may be separated into the determinants (see 9 and 10)

-0, x|4,6,6, 1. y 16,b,c,l, and $16,b,c,l.

unless the second and third of which are, by 8, equal

14,b,c,l-o;

(6) to zero. Hence *|4,b,c,! - Xboceli

in which case each unknown becomes , which

may have any value whatever. But the given lor, explicity (see 1),

equations may be written
s, b, c,

(i-1, 2, 3)
Kgb siis
a,b,c,
a,b,c,

any two of which will determine the ratios as baie

and

2. If these three equations form a conse Similarly, by starting with the determinants

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