FACTORIAIA 



FACTOBH "UY SYSTEM. 



The following are some of the OMM which are strictly intelligible 

 when U an integer, and in which the radical function remain* 

 intelligible when i fractional : 



lx - 



=/" 



/" 



-lt)=" y 

 " 



l-rfr 



r* *-'./, 



where i=p-f 



-r-'rfr 



here i=p-f 7. 



The second equation, which may be thua written, 



U the mart important The following are iU principal properties : 



1. T u finite when ti is positive, but is infinite when is or 

 negative. This draws a great distinction between it and !'' ; for 

 the general formula which connects factorials of positive an<l negative 

 exponent', with a common augment, U 



-"x (.r na)" = l 



IX which u infinite when M 0, diminishes from thence very rapidly. 

 becomes 1 when n = l, and thence further diminishes until 

 1 -401632. . ., when it is least, being then -8856082. . . . From thence it 

 begins to increase, being 1 again when -2, 1.2 whcnn = 3, 1.2.:: v. !. n 

 i = 4, and no on. 



2. The equation r( + l) = Tii. so obviously tme when n is an 

 integer, is always true ; giving alto 



l> + l)=n ( 1) ...... (-m) r( in) 



for every integer value of in. Hence a table of values which extends 

 through one unit is sufficient : and the most convenient interval is 

 that from i = l to n = 2. If, for instance, we wanted to calculate from 

 such a table the value of T (54) we should reduce it to 44 x 84 x 24 

 x 14 x F (14), and take T(l}) from the table. Similarly T, when i is 

 lens than unity, would be found from T (1 + )-=-. When u is very 

 email, r is 1-i-n nearly. 



3. The labour of calculating the table is much lessened by the 

 following equation : 



rnxr(l ) = 



which U true when n lies between and 1. One very useful result of 

 it is rjk = </*. 



4. There is a constant to be introduced, which we shall call 7, the 

 importance of which may in time compete with that of * and t. It is 

 the limit of the expression 1 + 2-' + 8 -l + .... + #- ' log .r, as .r is 

 increased without limit : it is also the value of 



7= -57721 50649015328606065. . . . 



ita value is 



5. If S. represent l-* + 2- > + 3-" + ad infinitum (See SUM- 

 MATION, for some of the values ; also ' Differential Calculus,' ' Library 

 of Useful Knowledge,' p. 554), the logarithm (Naperian) of T.r may be 

 found in either of tlie following ways : 



= 2 log (2) + (.? jj J log .r 



1 



Tli" 



l 



360r J1 



fi. A table of the values of comm. log. T(l + .r) is given by Legendrc 

 for every thousandth of a unit from ar'000 to .r=-f)9H : an abridg- 

 ment of this table, with means of completing it, is in the ' DitT. C'alc .' 

 ' Library of I'seful Knowledge,' p. 687. . This function IV is a 

 fundamental mode of expression for the result* of large classes 

 of definite integrals. [IBTF.ORAU, DKFINJTE.] 



7. The function r(n + l) is the value, or one of the values, of 

 ] I", and we hare 



.,'T +-''. , 7-ni 



i lioldn whenever (he pr-ond side in renl and finite. 



The student who desires to know more of the theory may ccm-ult 



Knuup's 'Analyse des Refractions A-ti, nii.|us,' Ktrasburg. 



4to., and the article ' FaetorieUes,' in the Supplemental i..r third i 

 v.ilnnie <if Montferrier'i- ore des Sciences Math 



Turin, 1840, 4to. Also the article ' Pacultat' in <! rum-it's Supplement 

 to Klugel's ' Worterbuch der Heinen Mathematik,' Leipzig, 1836, 

 J \"1-. Hvo. On the form r.r see Legendre'a well known 



es An Calcul Integral,' and that on Elliptic Fun. :i..n,. s.uuc 

 of the substance of these U in the treatise on the ' I'itl'. 

 Calculus', in the ' Library of I'seful Knowledge.' 



TORIES; r.\<T'nl!Y SYSTEM. The word factory has had 

 two different meanings. It formerly meant an establishment of mcr- 

 chant* and factors resident in foreign countries, who were 

 certain regulations adopted for their mutual support and assistance 

 against the undue encroachments or interference of the governn 

 the countries in which they resided. In modern times these factories 

 have, in a great measure, ceased to exist ; because of the greater degree 

 of security which merchants feel as regards both the justice of those 

 governments and the protection, when needed, of their own eonntry. 

 The Venetians, Genoese, Portuguese, Dutch, French, and Kngli.- 

 .\11 had establishments in the nature of factories. In China tl< 

 tugueae established a factory at Macao, and the English at Canton. In 

 most instances factories have at first obtained the privilege of trading, 

 and afterwards procured for the precinct assigned to them some 

 exemption from the jurisdiction of the native courts. In this M 

 things the supreme government of the country whose subject* have esta- 

 blished the factory prepare lawn for its control and administration, ami 

 treat it in fact as if it were its dependency, though the sovereignty of 

 the native government is undisputed. 



But in it* usual acceptation, the word factor)- has now a different 

 meaning. Modern legislation lias declared that a factory mean 

 building wherein steam, water, or other mechanical power is u 

 work any machinery employed in tlie manufacture of cotton, \vool. 

 hair, silk, flax, hemp, jute, or tow. 



What is called the Factory System owes its origin to the inventions 

 and skill of Arkwright ; and it is probable that but for the invention 

 of spinning machinery, and the consequent necessary aggregation of 

 large numbers of workmen in cotton-mills, the name would never 

 have been thus applied. It is in these mills that the factory - 

 has been brought to its highest state of perfection. The JHIV. 

 subdivision of employment according to strength and skill, and t 

 bringing to bear upon every distinct process exactly as much force an 

 is necessary, without waste, are the two great and valuable .-id vantages 

 of the factory system. The cotton-mills, and some of the circum- 

 stances connected with them, are noticed under COTTON MAM- 



PACTDRE. 



The legislature hag interfered to prevent children in factories from 

 being tasked beyond then- strength, to the permanent injury .if t heir- 

 constitutions. This abuse was the more to be apprehend, 

 large proportion of the children engaged in cotton-spinning are notdireotly 

 employed by the masters, but are under the control of the spinners a 

 highly paid class of workmen, whose earning* greatly depend \i]i! the 

 length of time during which they can keep their young assistant at 

 work. A parliamentary committee sat for the investigation of this 

 subject in 1832, and subsequently a commission was issued by the 

 crown for ascertaining, by examinations at the factories themselves, 

 the kind and degree of abuses that prevailed, and for suggesting the 

 proper remedies. In consequence of these inquiries, an act was passed 

 in 1833 for regulating factories. Attempts had been made in 1802, 

 1816, and 1831, to legislate for the protection of factory workers ; but 

 uly on a small scale. 



The Act of 1833 (3 & 4 Will. IV. c. 103) contains numerous details ; 

 but the chief matters that relate to the subject are the following : 

 1. After January 1, 1834, no person under 18 years of age shall IH> 

 allowed to work in the night, that is, between a quarter past eight r.M. 

 and half-past five A.M., in any cotton or other factory in which steam 

 or water, or any other mechanical power, is used to propel the macl i i 1 1 1 >ry , 

 except in lace factories. -. No person under 18 shall 1 

 more than 12 hours in one day, nor more than 60 in one \\. . k. 

 3. There shall be allowed in the course of every day not less 1 1 

 hours for meals to every person restricted to the perform..! 

 12 hours' work. 4. After January 1, 1884, no child shall be employ.-.! 

 under years of age, except in silk-mills. 5. After March 1 . 

 no child, except in silk-mills, shall be employed in any factory 

 than 48 hours in any one week, nor more than 9 hours in any day. v, h. 

 .-hall not be 11 years old; nor after March 1, 1835, who shall 

 12 years old; nor after March 1, 1886, who shall not be 18 years old .- 

 nnd these hours of work shall not be exceeded, even it" the child h.-i;- 

 worked during the day in more factories than one. 6. Chil.lt 

 young persons, whose hours of work are regulated, shall be entitled 

 to two holidays and eight half-holidays in the year. 7. Children. 

 whose hours of work are restricted to 9 hours a day, are not to be 

 employed without obtaining a certificate from a physician or surgeon, 

 certifying that they are of the ordinary strength and appear 

 children of the ages before mentioned, which certificate is tobe miei 

 signed by some inspector or justice. 8. The crown is to appoint. 

 during pleasure, four persons to be inspectors of factories, with 

 sive powers as magistrates, to examine the children employed in the 



