from the adjutant, which he communicates to his officers. Each company generally has two serjeants. SERJEANT AT ARMS, or MACE, an officer appointed to attend the person of the king; to arrest traitors, and such persons of quality as offend; and to attend the lord high steward when sitting in judgment on a traitor. Of these, by stat. 13 Rich. II. c. 6, there are not to be above thirty in the realm. There are ordinarily nine at court called the king's serjeants at arms, to distinguish them from others, who are created with great ceremony; the person kneeling before the king, his majesty lays the mace on his right shoulder, and says, Rise up serjeant at arms, and esquire for ever. They have besides a patent for the office, which they hold for life. They have their attendance in the presence-chamber, where the band of gentlemen pensioners wait; and, receiving the king at the door, they carry the maces before him to the chapel door, whilst the band of pensioners stand foremost, and make a lane for the king, as they also do when the king goes to the house of lords. There are four other serjeants at arms created in the same manner; one who attends the lord chancellor; a second the lord treasurer; a third the speaker of the house of commons; and a fourth the lord mayor of London on solemn occasions. They have a considerable share of the fees of honor, and travelling charges allowed them when in waiting, viz. five shillings per day when the court is within ten miles of London, and ten shillings when twenty miles from London. The places are in the lord chamberlain's gift. There are also serjeants of the mace of an inferior kind, who attend the mayor or other head officer of a corporation. SERJEANT AT LAW, or OF THE COIF, is the highest degree taken at the common law, as that of doctor is of the civil law; and, as these are supposed to be the most learned and experienced in the practice of the courts, there is one court appointed for them to plead in by themselves, which is the common pleas, where the common law of England is most strictly observed: but they are not restricted from pleading in any other court, where the judges, who cannot have that honor till they have taken the degree of serjeant at law, call them brothers. SERJEANT, COMMON, an officer in the city of London, who attends the lord mayor and court of aldermen on court days, and is in council with them on all occasions, within and without the precincts or liberties of the city. He is to take care of orphans' estates, either by taking account of them, or to sign their indentures, before their passing the lord mayor and court of aldermen: and he was likewise to let and manage the orphans' estate according to his judgment, to their best advantage. See RECORDer. SERGEANT MAJOR, a non-commissioned officer subordinate to the adjutant. SERGEANTY, serjentia, signifies, in law, a service that cannot be due by a tenant to any lord but the king. Though all tenures are turned into soccage, by 12 Car. II. c. 24, yet the honorary services of grand sergeantry still remain, being therein excepted. See KNIGHT-SERVICE. This word is the same with serjeantry; but it would puzzle an antiquarian to tell how or why such trifling variations of spelling have been introduced SERGESTES, a sailor in Eneas's fleet, from whom the Roman family of the Sergii claimed their descent. Virg. Æn. v. 121. SERGII, the surname of a patrician family of ancient Rome, which produced several great men and one great villain. See CATILINE, and SERGIUS. SERGINES, a town of France, in the department of the Yonne: nine miles north of Sens, and thirteen and a half south of Provins. SERGIPO DEL REY. See SEREGIPPE. SERGIUS CATILINA. See CATILINE. SERGIUS I., pope of Rome, was born at Palermo, and elected pope in.687. He died in 701, with a good reputation, after a reign of thirteen years and eight months. SERGIUS II. was a native of Rome; succeeded Gregory IV. in 844; and died in 847. SERGIUS III. was elected pope by the Romans in 898; but, the party of John IX. prevailing, he was driven from his seat, and did not recover it till A D. 905. He disgraced his dignity by his vices, and died in 911. SERGIUS IV. succeeded John XVIII. in 1009. He was humble and liberal minded. He died in 1112. SE'RIES, n. s. Fr. serie; Lat. series. Sequence; order. Draw out that antecedent, by reflecting briefly upon the text, as it lies in the series of the epistle. Ward of Infidelity. The chasms of the correspondence I cannot supply, having destroyed too many letters to preserve any series. Pope. This is the series of perpetual woe, Id. SERIES, in general, denotes a continual succession of things in the same order, and having the same relation or connexion with each other; in this sense we say, a series of emperors, kings, bishops, &c. In natural history, a series is used for an order or subdivision of some class of natural bodies; comprehending all such as are distinguished from the other bodies of that class, by certain characters which they possess in common, and which the rest of the bodies of that class have not SERIES, in music. See MUSIC. SERIES, in arithmetic and algebra, a rank or number of terms in succession, increasing or diminishing in some certain ratio or proportion. There are several kinds of series; as arithmetical, geometrical, infinite, &c. The two first of these are, however, more generally known or distinguished by the names of arithmetical and geometrical progression. These series have already been explained and illustrated in the article ALGEBRA, particularly the two first: it therefore only remains, in this place, to add a little to what has already been done to the last of these; viz. INFINITE SERIES. SERIES, HARMONIC, a series of terms formed in harmonical proportion. It has been observed in the article PROPORTION, that if three numbers vious, and it may be continued as follows: a. b. +r, &c. From inspection of the terms of a b a b 24and the ninth term is ·b'3a-2b' 4 a a b a b ·46' ab n—1.a-n- - 2. &C., If, in a series in harmonical proportion, a and b be two affirmative quantities, and such that a <b; then this series, which is positive at first, will become negative as soon as n— -2. b exceeds n-1. a. But, if ab, the series will converge; and, although produced to infinity, will not become negative. Let a and b be equal to 2 and 1 respectively; then this series becomes 7. 1.3.1, &c.; and, since if each term of an harmonical series be divided by the same quantity, the series will still be harmonical, therefore¦ §. . ., &c., is an harmonical series: whence the denominators of this series form a series of numbers in arithmetical progression; and conversely, the reciprocals of an arithmetical progression are in harmonical proportion. SERIES, INFINITE, is formed by dividing the numerator of a fraction by its denominator, being a compound quantity; or by extracting the root of a surd. An infinite series is either converging or diverging. A converging series is that in which the magnitude of the several terms gradually diminish; and a diverging series is that in which the successive terms increase in magnitude. The law of an infinite series is the order in which the terms are observed to proceed. This law is often easily discovered from a few of the first terms of the series; and then the series may be continued as far as may be thought necessary, without any farther division or evolution. An infinite series is obtained by division or evolution; but, as that method is very tedious, various other methods have been proposed for performing the this series, it appears that each term is formed by multiplying the preceding term by r; and hence it may be continued as far as may be thought necessary without continuing the di + Dr3 + E14, &c. B = a 1 D= ,E= = &c.; whence, by substitu a 1 tion, we have = + + + 3y2 + a 6 y3 3 y 3y2+ &c. a 2. Convert the quantity c2 + into an infinite series. Let the assumed series be A + By Cy2+ Dys, which multiplied by c2 + 2 cy-y2, gives c2=c2 A+ c2 By + c2 Cy2 + c2 Dy3, &c. +2cAy+2c By2+ 2c Cy3 -Ay1 By3. Now, by equating the coefficients of the homologous terms, we have c2 c2 A, c2 B + 2 c A = 5. Leta2 + be converted into an infinite o, c C + 2 c B—A = o, c2 D + 2 0 C − B = Now each term of the given series is to be compared with the correspondent terms in the first part of the above theorem; and, by substitution in the second, several terms of the required series will be obtained. Example 1.-What is the square of the series y-y+ y3—y1 + &c.? By comparing this with the general theorem, we find z=y, a = 1, b = o, c=— 1, d= o, g= 1, &c., and m = 2; whence y-y + y3 — y1\2—2y2 × (1−2 a x + c2x2 - 2ce. × (1—2 y2 + 3 y1 — 4 yo), &c., = y2 — 2 y1 + 3 y0 — 4 y3, &c. 2. Required the fourth power of the series 1 + x + x2 + x3, &c Here 1, a 1, b = 1, c = 1, d= 1, and m=4. +2e2gx) cere ), &c., = y2 ),&c.,= Then 1 + x + No2 + x3, &c.{^= 1 + 4 b x + 6 b2 x2 + 4 b3 x3 + ba x1, &c. + 4c +12 bc + 4 d 12 b2c SERIES, RECURRING, a series of which any term is formed by the addition of a certain number of preceding terms, multiplied or divided by any determinate numbers, whether positive or negative. Thus 2, 3, 19, 101, 543, 2917, 15671, &c., is a recurring series, each term of which is formed by the addition of the two preceding terms, the first of which being previously multiplied by the constant quantity 2 and the other by 5. Thus the third term 19 = 2 x 2 + 3 x 5; the fourth term 101 2 x 3 + 19 x 5, &c. The principal operation in a series of this nature is that of finding its sum.-For this purpose, the two first and two last terms of the series must be given, together with the constant multipliers. Let a, b, c, d, e, f, &c., be any number of terms of a series formed according to the above law, each successive term being equal to the sum of the products of the two preceding terms, the first being multiplied by the given quantity m, and the other by the given quantity n. Hence we shall have the following series of equations c=ma + nb, d= mb + nc, emc + nd, f= md + ne, &c. Then adding these equations, we obtain c + d + gtamxa+b+c+d+nxbtc tự te. Now the first member of this equation is the sum of all the terms except the two first; the quan- Let the sum of 3 v 160 55 + ) &c. SERIES, REVERSION OF, is the method of finding the value of the quantity whose several v3, &c., = 2 rv + powers are involved in a series, in terms of the quantity which is equal to the given series. In &c., whence = order to this, a series must be assumed, which being involved and substituted for the quantity equal to the series, and its powers, neglecting those terms whose powers exceed the highest power to which it is proposed to extend the series. Let it be required to revert the series ar +br2+cr+dxa + ex3, &c. = y; or, to find r in an infinite series expressed in the powers of y. Substitute y" for r, and the indices of the powers of y in the equation will be n, 2 n, 3 n, &c., and 1, therefore n= 1; and the differences are 0, 1, 2, 3, 4, 5, &c. Hence, in this case, the series to be assumed is Ay+By+Cys +Dy, &c., which being involved, and substituted for the respective powers of r, then we have ar = aAy+a By2+aCy3+aDy, &c. br+bAy2+2bABy3 +2bACy* { 1 35 * SERIES, SUMMATION OF, is the method of finding the sum of the terms of an infinite series produced to infinity, or the sum of any number of terms of such a series. The value of any arithmetical series, as 1a + 2a + 33 + 4.......... varies according as (n) the number of its terms varies: and, therefore, if it can be expressed in a general manner, it must be explicable by n, and its powers with determinate coefficients; and those powers, in this case, must be rational, or such whose indices are whole positive numbers; because the progression, being a whole number, cannot admit of surd quantities. Lastly, it will appear that the greatest of the said indices cannot exceed the common index of the series by more than unity: for, otherwise, when n is taken indefinitely great, the highest power of n would be indefinitely greater than the sum of all the rest of the terms. Thus the highest power of n in an expression exhibiting the value of 1a + 2a + 33 +43..n2, cannot be greater than n3; for 12+ 22 + 32 + 43...n is manifestly less than n3, or n2 + n2 + n2 +, &c., continued to n terms; but n', when n is indefinitely great, is indefi=nitely greater than n3, or any other inferior power of n, and therefore cannot enter into the equation. This being premised, the method of investigation may be as follows: =y } &c. +bB2y cr= +cA3y3+3A2By*, &c. dr= +dAy4, &c.) Whence, by comparing the homologous terms, b we have a Ayy; therefore A= B= a a ¿A') = 3 D &c., and consequently r= 5 b3-5abca3d a7 -, &c., = + 3 Example 1.-Required the sum of n terms of the series 1 + 2 + 3 + 4 + .....n. Let An3 + Bn be assumed, according to the foregoing observations, as a universal expression for the value of 1 + 2 + 3 + 4...n, where A and B represent unknown but determinate quantities. Therefore, since the equation is supposed to hold universally, whatsoever is the number of terms, it is evident that if the number of terms be increased by unity, or, which is the same thing, if n+ 1 be written therein instead of n, the equation will still subsist; and we shall have Ax n + 1)2 + B × n + 1 = 1 + 2 + 3 + 4......n +n-1. From which the first equation being subtracted, there remains A x n + l'—A n2 + B+n-1- Bn n+1; this contracted will be 2 An+ A + B = n + 1; whence we have 2 A- - 1x n+A+B−1 = 0: Wherefore by taking 2 A-10, and A+ B-10, we have A, and B = }; and consequently 2 r =, y=v, a = === 720 series 1+ 2+ 3, &c.? C= 1 288 6 1440,6 1 83 32 S |