PERIOD. 



PERIODS OF REVOLUTION. 



term usually applied to rectilinear figure* only, but without any par- 

 ticular reason for the restriction. 

 PERIOD, a none given to the recurring pwt of a CincruiTixa 



DtCtMAL. 



PKKIODIC ACID. poDrsi.] 



PERIODIC Ft The general con Detention ot periodic 



magnitude, that U, of magnitude which varie* in nuch manner ai to go 

 through stated cycle* of change*, each cycle being a reiteration of the 

 preceding one, ia the subject of TRIOOXOMXTRT. Though that science 

 derive* it* name from the* measurement of triangle* (ai geometry doe* 

 from that of the earth), iu reeouroee hare been cultivated and it* 



expanded until no definition abort of the preceding will 

 uinm* iu object or convey an idea of it* power*. 



In the article cited will therefore be explained the mode of measure- 

 ment applied to periodic magnitude : the present one i* intended for 

 nothing more than to pnint out a peculiarity of aome rlusrn of 

 ihehrsin function* which ha* procured for them the name of periodic 

 function*. 



The calculu* of Fcxcnox* consider* only form*, and the operation* 

 Decenary to convert one given form into another, or to aatUfy 

 equation* in which ome form* of operation are unknown. Periodic 

 functions are those which, performed any giren number of time* on a 

 rariable. reproduce the simple variable iUelf. Thus 1 x and - 7 

 are periodic functions of the second order, since 



l-(l-x)=z, -(-0-)=*. 



Again, 1 : (1 x) U a periodic function of the third order, since, if we 

 begin with x, and write 1 : (1 z) for x three time* in succession, we 

 end with x also 



* 



-1 



Periodic functions are remarkable in the calculus of functions for 

 the simplicity with which questions connected with them can be 

 solved, when compared with the difficulty of solution of cases in which 

 non-periodic function* enter. Their principal properties are here briefly 

 pointed out for further information, consult Babboge's ' Examples of 

 Functional Equations ' (appended to Peacock's ' Examples '), or the 

 article on the Calculu* of Functions contained in the ' Encyclopaedia 

 Metropolitans.' where further references will be found. 



Let +x be a function of x ; abbreviate <t"ta:) into Q~x ; <ti(<t>"x) into 

 f*x, Ac. : then, if $ be a periodic function of the nth order, <t>*x = x. 



Let r be one of the nth ROOTS of unity, then rx is the simplest 

 periodic function of the nth order ; but very simple ones of the form 

 (1 + la) : (c kx) may be obtained by making 



2(1 + cos) 



where I mean* the nth part of any multiple of four right angle*. For 

 e.if n = 4, may = 90', and co = 0, whence* (b> + ) : 2; 



2 + 2ot 



U a periodic function of the fourth order. 



Let 8z be any function of x whatsoever, and 8-'x its INVERSE 

 function, so that ee~'x = x; then if due be a periodic function, 

 0fe~'x is also periodic. Thus 1 x being periodic of the second 

 order, so are log (1 - f ), V(l - x 5 ), sin-' (1 - sin x), Ac. Ac. 



Let Bffc-'x be called a derivative of $x ; then if <fx and y-x be two 

 periodic function* of the nth ordur, cither can in an infinite number of 

 way* be made a derivative of the other. Thus one of the ways in which 

 1 : x i* a derivative of x ia -'". Let l,r, r>, . ...r"-' be all the 

 roots of unity, let ! be any function of x and y. and let r, be the same 

 function of fx and y-y, r, of q?s and <fy, Ac. From the equation 



P O -r r r, + r p, + + r i r_, = 0, 



And y in term* of x ; ayy=x. Then will <^y=9^c, or ^x = 8^0-' ,r; 

 that i*, Y-X u expressed u a derivative of fix. 



For example, let \z - x, t|>x = 1 z, be the periodic functions, 

 ot the second order ; then r = 1. Let P = < -f 6y, then the pre- 

 ceding equation become* 



ax + by 



( - 1) (- a.r 6 1 - y) - 0; 



6 - 2ox b-Uy 



^ -,-. -- 



The periodic function*, a* before observed, are those whose relations 

 are most easily obtained. For example, let ox, flz, yr, be given 

 functions of x, and Qx an unknown function, to be determined by the 

 equation 



yx<lax. 



If 02 be not periodic, there are two difficulties in the way, each most 

 requently insuperable : first, the determination of torn* one solution 

 of this equation; secondly, the determination of the 



function of this equation, or the solution of ifz = i|*. But when ox 

 i* periodic, both difficulties on be overcome and a general solution 

 given. Say *z = x, then rx i* any symmetrical function of x, or, 



o-> a*-'x; and if B,,, B,, Ac., be &r, tax. Ac., and r,, c,, Ac., be 



yx, ymjs, Ac., the general solution divides into two cases, in the first of 

 which the solution doe* not depend on the invariable function. 

 The moat general oate giro* 



*>a 



a, + c B, + CQ c, a, + . . + c c, . . c._ 



~ c u <', c 



In this case the preceding U the most complete and only solution- 

 But if c,, and B O . Ac., be *uch that the numerator and denominator of 

 the preceding both vanish, the general solution i* 



- 3) Co c, B, 



f (1 + 



+ C c, c, . 

 C, + ---- + C C, . . . C,_) fr. 



For demonstration and extension, seo the article cited in the 

 ' Enoyo. Metrop ,' H 192-198, and 235. 



I'KKIODS OK REVOLUTION. In the present article we umply 

 describe the names, commencements, lengths, and uses of those period* 

 which it is most requisite the reader should find distinctly explained 

 in a work of reference, premising some explanation of the way in which 

 companion of different periods give* new periods. 



By a period we mean a definite portion of time, beginning from a 

 given epoek, or trra* which, being repeated again and again, will serve 

 to divide all time subsequent to the epoch (or precedent, if the repe- 

 titions be also carried backwards from the epoch) into equal parts, for 

 the purposes of common reckoning and historical chronology. A period 

 is then a finite portion of time used for measurement, just a* a foot or 

 mile i* used for measurement of length. 



Periods may be divided into natural and artificial ; the former imme- 

 diately suggested by gome recurrence of astronomical phenomena ; the 

 latter arbitrarily chosen. Since, however, time cannot be preserved 

 and handed down aa if it were material, it in natural that the know- 

 ledge of artificial periods should be preserved by representing the 

 number of natural periods which they contain ; nor is it to be supposed 

 that artificial periods were ever invented in a perfectly arbitrary 

 manner, or were indeed ever anything more than convenient collections 

 of nntural periods. 



When one period is contained an exact number of times iu another, 

 each recommencement of the larger one i* also a recommencement of 

 the smaller one ; thus, the day being exactly twenty-four hours, if any 

 one day begin at the beginning of an hour, all days will do the same. 

 But if the smaller period be not a measure of the larger, a longer 

 period may be imagined, which in this article we shall call a cyJe, 

 consisting of the interval between the two nearest moment* at which 

 the smaller and larger periods begin together. Thus, a week of seven 

 day* and a month of thirty days give a cycle of seven months or thirty 

 weeks, these two period* being equal. If, however, the two period* 

 can be measured by a larger number of days, the cycle may be made 

 smaller ; thus, a month of 30 days and a year of 305 days, or a month 

 of 6 times five days, and a year of 73 times five days, would give a 

 cycle of 6 x 73 times five days, that is of 6 years or 73 months. 



\Vli.-n two natural period* are expressed by complicated fractions of 

 days, the method explained in FRACTIONS, CONTINUED, will serve to 

 show nearly how many of one period make up an exact number of the 

 other. Thus the tropical, or common year being 365^24224 days, and 

 the lunation being 29'630S9 days, both approximate, it appears that 

 86,524,224 lunar months would be 2,0.'i3,<i5U year* nearly. To reduce 

 this long cycle to others more convenient for use, and as accurate a* 

 the number of figures employed will permit, proceed as in the article 

 cited with the fraction 



2.953,059 



The quotients obtained are 12, 2, 1, 2, 1, 1, 17, Ac., at which 

 we stop, because the appearanne of so Urge a quotient a* 17, shows 

 that the result of the preceding quotients i* extremely near. The 

 successive approximations derived from the first ix quotients i 



! J? 



V.' 25 



_3 

 87 



99 



_ 

 186 



19 

 23fi 



Or 235 lunations make 19 years very nearly. 



The period in which all others are expressed is the d.iy, which is not. 

 a* many suppose, the simple time of revolution of the earth, but [DAY] 

 the average time between noon and noon. To distinguish it from 

 other day* it is called the nuan tolar day. 



The year, or the time between two venial equinoxes, is not a uniform 

 period, nor doea the average of one long period give prccitcly the aame 

 as another. [YEAR.] For chronological purposes, however, it in 

 uselen* to take account of this variation, and 365-2422114 days, the 



* Thwc terms are uwd ftvnonymotuly by mont chronologer* : tut some mean 

 by ra the whole ot the time which li moiurcd from the rpoch. The ear of the 

 reader will perhp bo fsmllltr with both of the phrc, " the Sth century 

 n/th Chrlitlin r," Mid " the Sth century <iftrr the Chrlitlun an." 



