117 



lilLI 



BINOMIAL THEOREM. 



14* 



document ; and at the present day a bill of sale is necessary to the 

 validity of all transfer* of shares in British ships, whether by way of 

 sale or of mortgage, and the instrument must be registered at the port 

 where the ship is registered. (17 ft 18 Viet. o. 104.) In general. bilU 

 of sale, being ex n termini founded on valuable consideration, are 

 available against the creditors of the seller. But by the operation of 

 the Bankruptcy laws, goods remaining with the consent of the true 

 owner in the order and disposition of the insolvent at the time of his 

 bankruptcy, are deemed to be the property of the latter, and pas* to 

 his acsignees to be distributed with the rest of his effects for the benefit 

 of his creditors. In all cases, a bill of sale of pennant rkatteU must be 

 registered, within twenty-one days after the making thereof, in the 

 judgment office of the Queen's Bench, otherwise it will, as against 

 assignees in bankruptcy or insolvency, or creditors, be null and void. 

 (17 ft 18 Viet. c. 36.) Moreover, in all cases such a deed may be set 

 aside on proof that it was a merely colourable and fraudulent expedient 

 for defeating the claim of band jule creditors, the courts of law being 

 little disposed to favour assignments of this kind, made secretly and 

 without the notoriety which attends the actual transfer of possession. 



BILLON, in coinage, is a composition of precious and base metal, 

 T-" ; "g of gold or silver alloyed with copper, in the mixture of 

 which the copper predominates. The word came to us from the 

 French. According to Boutteroue, in France, billon of gold was any 

 gold beneath the standard of twenty-two carats fine; and billon of 

 silver all below ten pennies fine. Boizard says that gold beneath the 

 standard as far as twelve carats fine, and silver to six pennies fine, were 

 properly base gold and base silver ; but that it was the mixture under 

 those quantities which made billon of gold and billon of silver, in con- 

 sequence of copper being the prevailing metal. Black money, or 

 billon, was struck in the mints of the English dominions in France, by 

 command of the kings of England, for the use of their French subjects. 

 Money of billon was common throughout France from about the year 

 1200. Hardies, authorised money of Edward the Black Prince, are 

 also found of similar mixture. It was probably one consideration with 

 Henry V II I. in coining base money, that it would circulate in France 

 to his advantage. Henry VIII. and Queen Elizabeth both coined base 

 money, approaching to billon, for the use of Ireland. 



Nothing precisely corresponding to billon is now coined by any of 

 the great European states ; although in some the abasement, or alloy, 

 of gold and silver coinage is carried to a much greater extent than in 

 England. 



BILLS OF MORTALITY, are returns of the deaths which occur 

 within a particular district, specifying the numbers that died of each 

 different disease, and showing, in decennial or smaller periods, the ages 

 at which decease took place. When the accuracy of these returns can 

 be depended upon, facts of great importance in their actual application 

 to the business of life may be deduced from them. From the mortuary 

 tables, commenced at Geneva in 1566, which have been continued until 

 the present time, it is ascertained that at the Reformation one-half of 

 the children born died within the sixth year; in the 17th century, not 

 till within the twelfth year ; and in the 18th century, not until within 

 the twenty-seventh year. Tables of this description, extending over a 

 long period, mark the progress of a country in wealth and happiness ; 

 and the share which political causes have had in producing the results 

 which they indicate, is a subject worthy of the highest consideration 

 of the statesman and politician. The Northampton Tables of Mortality, 

 also the Carlisle Tables, and the Swedish Tables, have served as the 

 chief basis on which annuities, life insurances, and other calculations 

 relating to the duration of human life, have been founded. The 

 London Bills were commenced after a great plague iu 1593. The 

 weekly bills were begun in 1603, after another visitation of still greater 

 severity. Imperfect as these documents are, there does not exist a 

 complete collection of them, not even in the British Museum. In 

 London, the Bills of Mortality included the City of London, the City 

 and Liberties of Westminster, the Borough of Southwark, and thirty- 

 four out-parishes in Middlesex and Surrey, the whole containing a 

 population of 1,178,374 in 1831 ; but they did not contain St. Luke's, 

 Chelsea, population, 82,871 ; Kensington, 20,902 ; St. Mary-le-bone, 

 122,206 ; Paddington, 14,540; and St Pancras, 103,548; total, 293,567. 

 In the year in which the census was taken (1831) the number of deaths 

 published in the annual Bill of Mortality was 25,337, or 1 in 46, on a 

 population, as above stated, ef 1,178,874. 



In 1837 this system, if system it could be called, was wholly abolished 

 by the passing of the Registration Act of the 6 ft 7 Will. IV. c. 86, 

 by which returns of all the births, deaths, and marriages, in England 

 and Wales were secured, and which was extended to Scotland in 1854, 

 by the 17 ft 18 Viet. c. 80. [REGISTRATION ov BIRTHS, DEATHS, 

 AND MARRIAGES.] 



BINARY CoMPOrNDS. [NOMEXCLATURE, CHEMICAL.] 



I'.INAIIV TilKORY. [SALTS.] 



I'.INKTHIDE OF TIN. [OROAXOiiETALLic BODIES.] 



BINITROBENZOIC ACID. [BESZOIC ACID.] 



BIN1TROBENZOLE, ,it. [BKXZOLK.] 



HIXITKiiXAIMITHAI.IN, IXrtUnnapkthuUn. [XAI-IITIIAMX.] 



I!IN\A(,'LK, an article used on board ship which contains the 

 compass. It is placed next the steersman, and i* often divided into com- 

 partments for containing an hour-glass and a lamp. In order that the 

 compass may remain unaffected by any local cause, the binnacle is not 



nit together with nails or any iron work. The binnacle has of late 

 wcome an object of great consideration, from iu being made to contain 

 arrangements of correcting magnets and soft iron, on principles so ably 

 worked out by Professor Airy. [LOCAL ATTRACTION.} Bittacle, being 

 an abbreviation of the French word habtlarlr, a small habitation, was 

 the name formerly given to this article, and it is so called in Johnson's 

 Dictionary ; ' but it is now written binnacle. 



BINOMIAL, in algebra, means an expression which contains two 

 terms, such as 



a + 6 b ex a*xpy. 



Any expression may be considered and used as a binomial in any sense 

 iu which it may be said to contain two terms : thus, 



a + b + cx ex 

 when put in the form 



(a + 4) + (c e)x 



is a binomial, the terms of which are a + 4 and (ce) x. 



I'.IVOMIAL EQUATION. [EQUATION, BINOMIXAL.] 



BINOMIAL THEOREM, by far the most important thcor. 

 common algebra, first announced by Newton, as will presently appear. 

 It is frequently called on the Continent the bixome tie .Vfirton, and is 

 often said, but wrongly, to have been engraved on his torn'' iu 

 Westminster Abbey. In explaining this theorem, we shall << 

 ourselves as writing for those who have already such a knowledge of 

 algebra as will enable them easily to recognise the various expressions 

 of which we make use. 



The binomial theorem, coupled with those preceding theorems from 

 which it springs, is as follows : 



(1.) If a be denoted by a 1 , aa by a*, and aaa by a*, &c., then 



o-xo'=o"+ = a*-*(m>n). 

 a* 



(2.) The equations in (1.) will hold good when the symbol a." in 

 considered, provided that a always signifies unify. 



(3.) The equations in (1.) will hold good when negative exponent* 

 are employed, provided that 



i 



a~* means cr 

 a 



(4.) The equations in (1.) will hold good when fractional exponents 

 are employed, provided that 



a* means the square rout of a 



and also that 



a* cube root of a 



a* fourth root of a, &c. 



o* means (a-)* the cube root of a- 



o i, (o 4 ) the seventh root of o* 

 i_ 



<" a (*)" the nth root of a". 



(5.) Binomial Theorem. In all the preceding cases, that is, whether 

 n be whole or fractional, positive or negative, 



^l x> + n 2 i? 

 2 23 



_ n 1 n 2 n 3 

 + _ _ -- 



the preceding being a series of an infinite number of terms iu all cases, 

 except only where n is a positive whole number. Thej<th term of the 

 preceding expression is 



= 1+ tu: + 



. 

 2 3 



which expresses any term after the second 



(6.) The preceding series is convergent, whatever may be the value 

 of n, whenever x is less than 1. If .c be greater than 1, it U always 

 divergent ; but the series remaining after any term may be expressed 

 in a finite form, as follows : Let v,, v,, v,,, ftc., represent the several 

 terms of the preceding series, then all the terms after thepth term are 

 an algebraical development of a term of the fium 



where 9 is a function of x, the arithmetical value of which is less than 

 unity, so that 



(!+*) =v, + V,+ .... -fv, +v,+i (1 + flx)*-* 



n 1 



v, = 1 v, = HJC v = n - 1 x 1 , &c. 



The preceding theorem, though theoretically necessary to those who 

 do not allow the UPC of dhngent series, is of no practical uso in tin- 

 determination of (1+x)", since the determination of itself is the 

 more difficult problem of the two. 



We shall now give the early history of this theorem, with some 

 remarks upon its demonstration. 



