SERAPIHXE. 



SERIES. 



UM 4th year of Darius. All the aacred bull* were not buried with the 

 max honour*. tho*e mot carefully mnhalmnd having their princi]>al 

 bone* placed in a wooden coffin iomde the granite sarcophagus ; others 

 were out into a hollowed place in the rooky pavement, and covered 

 with a men flagstone. The tablet* found consisted chiefly of hiero- 

 glyphic*! inacriptions intermixed with demotic, commencing with the 

 lith year of Darius, and consisted of two clmei : the sepulchral, or 

 tombstones of the bull, on which waa inscribed the date of the death 

 of the acred bull, that of hi* birth, and of hi. installation at Memphis, 

 and the ace which he had reached ; and votive tablet* dedicated by 

 individual* to the departed Apia for the usual benefits supposed to be 

 accorded br the god*. About 1200 of then tablets were found, and 

 they hare thrown great light upon the chronological period and suc- 

 cession of the monarch* of the 21st and 22nd dynasty. Altogether, 

 7000 object* wen found, 8000 of which referred to the worship oi 

 Apia, a number by no means remarkable when it is considered that 

 Ptolemy L apent about 10.000/. on the funeral of an Apia. The 

 sepulchres, were of bulls which died in the following reigns Ame- 

 aophii III., Amenankhut, Horus, and Khuenaten, of the 18th dynasty ; 

 ot Setho* I. and Ramesea II., three of which happened in the 16th, 

 26th, and 30th yean of the last-named monarch, of the 19th dynasty ; 

 of RamesesIII , V.. VIM., XIV., of the 20th dynasty; of Osorchon II., 

 Takelothis I., Sheshonk III., IV., of the 22nd dynasty ; of Boochoris, 

 of the 24th dynasty ; of Sabaco and Tirhaka. of the 25th dynasty ; of 

 Psammetichus I., II., and Aahmes, of the 26th dynasty ; of Cambyses, 

 Dariiw, and Khabaoh or Smerdis, of the 27th dynasty; of Nethther- 

 bcbi, of the 30th dynasty. The tablet* subsequent to thin reign, 

 chiefly demotic, consist of votive inscriptions in honour of Apis, and 

 they gave the following dates for the appearance of the Apis : B.C. 258, 

 in the 32nd year of Ptolemy Philodelphus ; B.C. 231, the 20th year of 

 Euergetes; B.C. 210, the 14th year of Ptolemy Philopator ; B.C. 185 

 and 165, in the reign of Ptolemy Philometor; B.C. 142, the 13th 

 yearof Ptolemy Euergetes II.; and B.C. 117, in the reign of Ptolemy 

 Euergetes II. The temple of JEeculspius was also partially uncovered 

 by M. Mariette, but tradition hod assigned the spot to be the prison of 

 Joseph, and the work could not be carried on. 



(Marietta, Aug. Choix de Monuments dcrourerts pendant Ic de place- 

 ment du Serapatm de Memphis, 4to, Paris, 1856 ; Memaire sur la mire 

 fApit, 4to, Paris, 1856; L'Athenaum Francois, 1855, 1856.) 



IIXE, a musical instrument of the keyed kind. It may be 

 described as a small organ, in which short, thin, and narrow steel bars, 

 or springs, put into vibratory motion by means of a bellows acted on 

 by the foot, are used instead of pipes. It was one of the predecessors 

 of the HARMONIUM, which now takes tho place of such instruments as 

 are not portable. The portable forms are represented by the CON- 

 CERTINA. See also ACCORDION. 

 SEKAPIS. [SKHAI-EUM.! 



SERENADE, a word adopted from the French iMnade, which is 



om the Italian and Spanish icrenata, a term formed from the 



Latin iercH<it, clear, serene. A serenade is properly music performed 



in the open air on a serene night, but is generally restricted to a 



U performance given at night by a lover to his mistress under 



The giving of serenades is little practised except by the 



Spanish and Italians, who generally on these occasions use the guitar 



as an accompaniment to their songs. 



SERIES. The mathematical meaning of the word series is, a set of 

 inns, finite or infinite in number, connected together by addition or 

 subtraction, and formed upon some distinct law. If it had been the 

 plan of this work to write treatises on the various branches of pure 

 mathematics, the present article would have been brief, and abounding 

 in references to the articles on algebra and the differential calculus 

 the most important results of which are expressed in series ; but in a 

 work which, without entering into such full details, professes to 

 Furnmh references to the most important detached doctrines of the 



ees, the present article must extend to some length 

 Series may be either finite or infinite in the number of their terms. 



finite series, such for instance as x terms of 1 + 2 -f 3 +.. 

 the only question of importance which generally arises with respect to 

 is, how to express the sum as a function of the indefinite 

 number ofterms, x. On this point we refer to the articles INTE- 

 GRATION, FINITE, and SUMMATIOK : it is with the doctrine of infinite 

 that the mathematician is more particularly concerned in tho 

 t article. Again, as to the manner in which the differential 

 t0 devel P ment of functions in aerie*, we refer 



i- * ^ numb ? r l termB ""y ^ either P"^ nume - 



rical, as 1+2 + 8 + 4+ ..... , in which the symbol + ... or + Ac 

 means that the series U to be carried on for ever, the law of formation 



^ ^TSf i' ng CODt ! nued throu sn ^ the unwritten ones ; 

 ntata literal expressions with an obvious law of formation, 

 + . . . . A series of the latter class is reduced to one 

 of former so soon a, any definite value i* given to the letters it 



1 ^ fl f ito .. leri ! . "7 **> ""her convergent or divergent, as 

 explained in the article CONVERGENT. The varioua tests there ex- 

 plained will perhaps serve to settle this point as to the greater number 



sene. octuaUy employed ; but the following ( Diff. Cak' L b 

 Useful KnowL,' pp. 236, 326 ; we shall refer to this work k Uhe sequel 



under" the letters D. C.) will leave no doubtful cane, though its appli- 

 cation may sometimes be troublesome. 



Let +x be the *th term of a series ^l + i^2 + ^3 + (thus n 1 -' 



is the *th term of I + o + o'+ ; .ro 1 of l + 2a + 3a+ ), 



and let p a = x^x : <l>x, <ff x being the differential coefficient 

 Let a be the limit of P when x increases without limit ; then if a a be 

 greater than 1, the series is convergent ; if o be less than 1 (negative 

 quantity included), divergent ; if = 1, doubtful. 



In the doubtful cose of the preceding, let P,=log x (P O 1), and 

 let o, be its limit when x increases without limit. Then if a, be >l, 

 the series is convergent; if <1, divergent ; if =1, doubtful In this 

 doubtful case examine r, = log log x (p, 1), of which let the limit bo 

 o,. Then if a, > 1, the series is .'convergent ; if < 1, divergent ; if = 1, 

 doubtful ; and so on. In brief, take the set of quantities 



f* 



PU= ~ x Jx' v i = } 8 x ( r o~l), P,=loglogar(P 1 -l), 

 r, = log log log x (p, - 1), 4c. &c. ; 



make x infinite ; then, according as the first of these which differs from 

 unity is greater than or less than unity, the series is convergent or 

 divergent. If it be more convenient to write 1 : <px, instead of . 

 the ;rth term of the series, then p (1 must be x<f>'x : <t>x, instead of 

 x<l/x : tyx. Nor need ^x be the xth term ; it may stand for the 

 (.c + n)th term, n being constant 



By the symbol Sf* is here meant the series <f>x + <t>(x + 1 ) + <f(x + 2) + 



; but when a number is written beneath s, 03 in s,, it indicates the 



value of x in the first term of the series. Thus B t x stands for 

 4 + 5 + 6 +...., s. log x stands for log a + log (a + 1) + .... Some 

 such abbreviation is most wanted in an article of reference, in which 

 compression is desirable ; but the student should write his series at 

 more length until he is well accustomed to them. 



A divergent series is, arithmetically speaking, infinite ; that is, the 

 quantity acquired by summing its terms may be made greater than 

 any quantity agreed on at the beginning of the process. Such ii 



evidently the case with 1 + 2 + 4+ , or S 2'. Nevertheless, as 



every algebraist knows, such series have been frequently used as the 

 representatives of finite quantities. It was usual to admit such series 

 without hesitation ; but of late years many of the continental mathe- 

 maticians have declared against divergent series altogether, and have 

 asserted instances in which the use of them leads to false results. 

 Those of a contrary opinion have replied to the instances, and have 

 argued from general principles in favour of retaining divergent series. 

 Our own opinion is, that the instances have arisen from a misunder- 

 standing or misuse of the series employed, though sufficient to show 

 that divergent series should be very carefully handled ; but that, on 

 the other hand, no perfectly general and indisputable right to the UBO 

 of these series has been established d priori. They appear always to 

 lead to true results when properly used, but no demonstration has 

 been given that they must always do so. 



Before, however, we proceed to reason upon them, we must distinctly 

 understand what we mean by an infinite series. Some persons cannot 

 imagine an infinite series, except by means of successions of finite 

 terms : thus they have no other idea of 1+2 + 4 + 8 + ..., except as 

 something of which the conception is a pure result of the successive 

 consideration of 1, 1 + 2, 1 + 2 + 4, 1 + 2 + 4 + 8, &c. If they can get no 

 further than this, that is, if at no stage of their contemplation can they 

 treat 1 + 2 + 4 + . . . as anything more than carried to some enormous 

 number of terms, with a right to carry it further ; we can then concede 

 to them the right to object, in the manner described, to the use of a 

 divergent series, though we think it possible that even in this case an 

 answer might be given to the objection. But if there be any who can 

 with us carry their notions further, and treat the series as absolutely 

 endless, in the some manner as we are obliged to conceive time and 

 space to be absolutely endless, looking upon the result not as to its 

 arithmetical value, but as to its algebraical form and capability of being 

 the object of algebraical operations we then think that we have those 

 with whom the question of divergent series can be argued on something 

 more like a basis of demonstration. They may arrive at the final idea by 

 means of the successions which the first class of thinkers say must end 

 somewhere ; but they answer, that this is no more true than that space 

 must end somewhere : if it be granted that we are capable of conceiving 

 a straight line extended without limit, with equal ports set off throughout 

 its total infinite length, it must equally be granted that we might 

 suppose one term of a scries written at each and every point of sub- 

 divUion. To this issue tho question might be brought, namely the 

 alternative of allowing the conception of the infinite series or of 

 denying that of the infinite straight line. And it must be remarked 

 that the considerations by which we limited the use of the word 

 INFINITE in that article do not apply here, for we are not reasoning 

 upon any supposed * attainment of the other end of the straight line- 

 but upon ideas derived from a process of successions carried on during 

 such attempt as we can make in our thoughts towards that attainment. 

 This being premised, let us now consider the series 1 + o + o* + o 3 + . . 



* Indeed it is only a phraseological attainment of infinite magnitude which 

 nsed in the article cited : when we tay that a = b if s be infinite, we mean 

 that a and never cease to approach each other to long as the value of z 



