COMBAT, 8TXGLE. 



COMBINATIONS AND PERMUTATIONS. 



which b a ohiMl equal in breadth to UM length of tho intended teeth 

 erf the comb, in brought down with nifllcient fore* to cut completely 

 through the bom or tortoiseshell. As the teeth we not require.) to be 

 perfect!? parallel, but in a alight degree wedge-duped, the successive 

 euta of the ohiMl murt incline a Uttk obliquely, first in one direction, 

 and then in the other, for which provfadon U mad* by an arrangement 

 for alternately changing the position of the eatter during itadenanl 

 A pair of very narrow chud* mounted at right anglei with, and at the 

 end* of, the long cutter, serve, by their alternate deaceot, to connect 

 each pair of cuta at their converging extremities, so as to detach tho 

 end* of the teeth. In Mr. Rogers'* machine for the lame purpose, 

 everything U effected by the continuous turning of a winch, which, >>y 

 mean* of a crank, works a double cutter, the two chuck of which are 

 oapabU of adjtMtment according to the sue of the teeth, while the 

 crew for moving the bed or carriage w turned by mean* of a cog- 

 wheel upon iu azb, working into another on the axis of the winch. 

 The extent of motion between each operation of the cutter U regulated 

 by varying the relative aim of theM oog-wheels. 



Among recent novelties in comb-making is Mr. Griffith's curious 

 patent, in 1852, for "galvanic combe," of which the teeth arc alter- 

 nately of copper and sine, while the handle is hollowed into a chamber 

 for containing a roll of flannel moistoned in acid solution. The in- 

 ventor seems to expect a beneficial galvanic action by combing the 

 hair with thin apparatus. A more promising novelty is the appli- 

 cation of the very tough substance called vulcanised india-rubber 

 [CAOUTCHOUC MANt-FAC-mirs] as a material for combs. 



COMBAT, SINGLE. JDoiL.] 



COMBINATION LAWS. Tho IAWS known by this name were 

 repealed in 1824. Till then any combination of any two or more 

 masters, or of any two or more workmen, to lower or raise wages, or to 

 increase or diminish the number of hours of work, or quantity of work 

 to be done, was punishable at common law as a misdemeanour : ami 

 there were also thirty-five statute* in existence, most of them applying 

 to particular trades, prohibiting combinations of workmen ai?iin*t 

 masters. The Act paused in 1824 (5 Gee, IV. c. 05) repealed all the 

 statute and common law against combinations of masters and of work- 

 men ; provided a summary mode of conviction, and a punishment not 

 exceeding two months' imprisonment for violent interference with 

 workmen or masters, and for combinations for violent interference ; 

 and contained a proviso with regard to combinations for violent inter- 

 ference, that no law in force with regard to them xhotiM be altered or 

 affected by the Act. But all the common law against combinations 

 being repealed by tho Act, this proviso was considered as of no force ; 

 and the Act also went Ipryoncl tin- intentions of theframers in legalising 

 combinations unattended with violence for the purpose of controlling 

 masters in the mode of carrying on their trades and manufactures, as 

 well as peaceable combinations to procure advance of wages or reduction 

 of hours of work. The Act was passed after an inquiry into the subject 

 by a committee presided over by Mr. Hume, which reported to the 

 House the following among other resolutions : 



" That the masters have often united and combined to lower the 

 rates of their workmen's wages, as well as to resist a demand for an 

 increase, and to regulate their hours of working, and sometimes to 

 discharge their workmen who would not consent to the conditions 

 offered to them ; which have been followed by suspension of work, 

 riotous proceedings, and acts of violence. 



" That prosecutions have frequently been carried on under the 

 statute and the common law against the workmen, and many -of them 

 have suffered different periods of imprisonment for combining and 

 conspiring to raise their wages, or to resist their reduction, and to 

 regulate their hours of working. 



" That several instances have been stated to the committee of prose- 

 cutions against masters for combining to lower wages, and to regulate 

 the hours of working ; but no instance has been adduced of any master 

 having been puni><hed for that offence. 



" That it is tho opinion of this committee that masters and workmen 

 should be free from such restrictions an regard the rate of wages and 

 the hours of working, and lie left at perfect liberty to make such 

 agreements as they may mutually think proper. 



" That therefore the statute laws which interfere in these particulars 

 between masters and workmen should be repealed ; and also that the 

 common law, under which a peaceable meeting of masters or workmen 

 may be prosecuted as a conspiracy, should be alter. 



Immediately after the passing of this Act a number of widely 

 organised and formidable combinations arose In various trades and 

 manufactures for the purpose of controlling the masters as to the way 

 in which they should conduct their business ; and the extent to which 

 the Act had repealed the common law being doubtful, and the Act 

 having clearly gone beyond the resolutions on which it was grounded 

 in legalising combinations, Mr. Huskimon, then President of the Board 

 of Trade, moved early in the session of 1825 for a committee to con- 

 sider the effects of the Act 5 Oeo. IV. c. 95 ; and a committee was 

 appointed with Mr. (afterwards Lord) Wallace, then Vioe-President of 

 the Board of Trade, for its chairman. This committee recommended 

 the repeal of the Act of the previous session, and tho enactment of 

 another ; and in consequence of their recommendation the U. I V. 

 c. 129, was pawed, which is the Act now in force relative to combi- 

 nations. 



This Act repealed the 6 Geo. IV. c. 95, and all the statutes which 

 that Act had repealed. It relieved from all prosecution and punish- 

 ment persons meeting solely to consult upon rate of wagra or hour- of 

 work, or entering into any agreement, verbal or written, on these 

 points. And it provided a punishment of not more than three months 

 imprisonment, with or without hard labour, for any one using violence, 

 or threats to make a workman leave his hiring, or return work un- 

 finished, or refuse to accept work, or belong to any club, or contribute 

 to a common fund, or pay any fine for not belonging to a club, or con- 

 tributing to a common fund, or refusing to conform to any rules made 

 for advance of wages or lessening of the hours of work, or regulations of 

 the mode of carrying on any business, and for any one using violence 

 to make any master alter his mode of carrying on his business. 



By the Act 6 Geo. IV. o. 129, therefore combinations of masters and 

 workmen to settle as to rate of wages and hours of labour are made 

 legal and freed from all punishment ; but the common law remains a* 

 it was as to combinations for otherwise controlling masters. Although, 

 therefore, workmen may conspire together not themselves to work 

 under certain wages, they must carry out their object by lawful means, 

 and cannot intimidate or prevent masters from employing, or workmen 

 from taking employment, at any wages those other workmen may 

 agree for. (Reg. t>. Rowlands, 2 Den. C. C. 864.) 



By 9 Geo. IV. o. SI, assaulte in pursuance of a combination < 

 the rate of wage* are made punishable by imprisonment and hard 

 labour. 



COMBINATIONS AND PERMUTATIONS. By the word ..,,,1 

 bination is usually meant any selection which can be made out of a 

 number of different object* without reference to tho order in wlii.-li 

 they ore placed ; while by a permutation is meant a conii'in.ni. .n in 

 which different orders of position are to be considered as constit 

 specific difference. Thus abed, acbd, dacb, are all the same com- 

 bination of four out of the alphabet, but different permutations of four. 

 The investigation of questions relating to combinations, &c., i the 

 principal mathematical part of tho theory of probabilities, and was first 

 considered in detail, with reference to that science, by James Bernoulli! 

 and Montmort (see Library of Useful Knowledge: 'Probability'); 

 but the common rules had previously found their way into an tin 

 treatises. The enormous number of different amngementa of which 

 objects are susceptible, even when their number is not large, drew 

 early attention to the subject. We shall give some of the most winkle 

 rules, and a help to calculation for high numbers. 



I. The number of permutations having x in each, which can be made 

 out of X things, is the product of x terms of the series, 



X,X 1, X 2, X 3, ftc. 



Thus, out of 10 tilings, there are 10 x 9, or 90 permutations < 

 10 x 9 x 8, or 720 permutations of three; 10 x 9 x 8 x ", ... 

 permutations of four; and so on. Finally, the number of different 

 arrangements which the whole ten will admit of, nay the number of 

 change* which can be rung on ton bells, is 



10. 9. 8. 7. 6. 6. 4. 3. 2. 1. or 8,628,800. 



II. When the whole number of things, X, contains a which arc 

 alike of one sort, 6 which ore alike of another sort, &c., the total num- 

 ber of arrangements of the whole is not the product of X, X 1, tx., 

 down to 1, but that product divided by the product of 1, 2, 3. ... up 

 to a, then by that of 1, 2, 3. ... up to b, Ac. Thin result can be most 

 easily formed by striking out common factors from the numcratora and 

 denominators. 



III. In the last case, the number of permutations of x out of X 

 being required, no simple rule can bo given, but each casein 

 solved by itself. For instance, how many permutations of three . 

 formed out of 



n a a b b e. 



(I.) All being different, 3. 2. 1. or 0. (2.) Where a is rv; 

 twice, wo have 6. (8.) Where a is repeated three times, one only. 

 (4.) Where 6 is repeated twice, we have 6. In all, 1!>. 



IV. Tho number of combinations of x things out of X, all 

 different, is 



prod, of x terms of X, X 1, X 2, ftc. 



divided by 

 prod, of x terms of 1, 2, 8, &c. 



Thus out of 10 things, the number of combinations of four in in. it. 

 8. 7. divided by 1. 2. 8. 4, or 210. The best way of arriving at this 

 result is by destroying common factors, which shows it to be 5. 8. 2. 7. 

 Observe also that we may shorten this process, when x is great 

 the half of X, by finding out, not how many selections can be taken, 

 but how many remainders can bo left. Thus the number of co> 

 lions of u'.'i "lit of 30, is the same as the number of conibiinti.nm of fl, 

 for 26 can only be taken in as many ways as 5 can 1* 



V. The number of combinations of x things out of X, any rej" 

 being slim. 



prod, of . \ I- 1, X + 2, Ac. 



divided by 

 1. of X terms of 1, 2, 3, &c. 



VI. The number of ways in which n places may be filled up f. 



