

DIF; 



Ut*u'.. t>f an inch), a piece of iron, of the same diameter u the 

 iii.l iK-rbnp* half an inch thick, U to be covervd with wax on 



: i ...'.. ' >' 







tU surface. The deposited tueUl [of the galvanic trough] will now be 

 thrown down between the two ; and U the strength of the battery bo 

 carefully regubu-d, they will be united into one whole piece. All the 

 wax may now be remove;!, and the edge* trimmed with a file. We 

 shall thiu hare one of the die* in which to eaat the button ; the other 

 mar be "<t. in the lauie manner. Thu method U applicable to the 

 ^.tia- 0| die* for casting all kiiuU of ornament* in soft metal, such u 

 teapot knob*, handle* ana feet -i cruet frame*; also die* for stamping 

 oft material*. a* leather *hot pouchis*, bookbinder*' work, &c. A* the 

 temper of the metal deposited depends entirely upon the intensity . i the 

 battery-current, I have no doubt that, if this were carefully regulated, 

 die* might be made fit fr stamping Britannia metal or copper. The 

 great recommendation of the process in its extreme cheapness." 



Dl'ESIS, in ancient music (litar.s, ilii-isiun). The Qreelu divided a 

 tone into a major and minor semitone ; the greater was called an 

 apotome [APOTOME], the lesser a liuiiua, or Diesis: to the difference 

 between the two the name of comma [COMMA] was given. But it mu.-t 

 here, once for all, be observed, that the Greek writer*, if we really 

 enter into their meaning, differ much in their definitions of musical 

 inetrvala, and that the moderns are no less at variance in their inter- 

 pretation of many ancient terms of the art. 



DIETHYLAMIXE. [OHUA.MC BASLS ] 



D1ETHYLAXCYLAMIXE. [ORGANIC BASES.] 



DIETHYLAX1L1XE. Synonymous with dictAylpJc.HytainiHt. [Axi- 



un.1 



mETHYLCHLORASILIXE (X(C,,H.C1(C.H 5 ),). A derivative 

 from chloraniline. [AMLI.NE.J 



D1ETHYLCOX1NE . An organic base formed by re- 



placing two equivalents of hydrogen in conine by two of ethyl. 

 [Covon.1 



CHA 



An organic body consist- 



43, 47, 53, 61. 71, 83, Ice., are the value*, when .r is successively made 

 equal to 1, 2, 3, 4, 5, 6, Ac. f This problem i* indeterminate, with 

 respect to common algebra merely, unless we can ascertain a law by 

 which the aerie* can be continued 



able to assume; it then becomes determinate with re--: 

 expression* of common algebra only, but is again indeterminate if we 

 are allowed to assume the transcendental expressions of trigonometry 

 or the integral calculus. Confining ourselves to the expressions of 

 common algebra, we may detect any law which prevails throughout 

 the whole series by taking the difference between each term and the 

 next, and thus forming a new series ; and then by repeating this pro- 

 cess again and again until the series so formed presents an easily 

 perceptible law. Thus in the preceding case we have 



(iiven series 43 47 53 61 71 83.. 



fC.HA 

 X i. C.U., I 

 Icy / 



DIETHYL-CYANAMIDE ( X 



ing of cyanamid in which two equivalents of hydrogen are replaced 

 by two of the radical ethyl. [CYANAMIDZ.] 



DIETHYLCYAXl'lUC ACID (C^HJ^O,,, HO). A crystalline 

 btained as a secondary product in the preparation of cyanuric 

 ether. Fused with potash it yields ethylamine. 



D1ETHYLIXE (C 14 H, (),,). This organic compound is formed by 

 the juxtaposition of one atom of glycerine with two atoms of alcohol, 

 and the abstraction of four atoms of water. 



C.II.O, 4- 20,11,0, 4HO = C lt U l( 04 



.1 ^, First differences 



. ^, 

 c 



Second differences 



4 U 



2 



8 10 12 



2 2 



(Jliccrine. Alcohol. 



Uit-thjline. 



prepared by heating a mixture of glycerine, bromide of ethyl, 

 and potash to 212" Falir. for about SO hours, then separating the upper 

 layer of resulting liquid from the lower, and distilling the former. 

 That portion of the distillate which passe* at 37tS Fahr. is pure die- 

 tlivliuv. This body is a colourless and transparent liquid, possessing 

 slight ethereal odour and a specific gravity of '92. It is but slightly 

 soluble in water, lli-.ite.l with quicklime it disengages acroleiu. 



DIEfHYLMECOXIC ACID, [iluoosic ACID.] 



DIETHY LUX AMIDE. [OxAiuuE.J 



DIETHYLl'HEX YLAM1XE. [AMUSE.] 



UIETUYLPHUSI'HOKIU ACID. [PuospHOBic ACID.] 



DlETHYL-TOLUlDIXE. [ToLUiuixE.] 



DIETUYL-UKEA. [L*BEA.J 



DIET. [FooD.] 



D1FFAKKLAHOX. [MABBIAGE.] 



DIFFERENCE, the excess of one quantity over another. This 

 fundamental meaning of the term is almost lost in the high 

 of mathematics, from the association of it with a methodised theory, 

 derived from the consideration of the differences presented by 

 successive quantities which follow a regular law. It is, therefore, a 

 very wide branch of pure mathematics which must be considered 

 under this term ; namely, the method of calculus of difference-. And 

 the connection of this subject with the differential calculus (the results 

 of the Utter being, in one point of view, particular cases of the former) 

 renders it impossible to treat of the two with that perfect separation 

 which the alphabetical arrangement of a work like the present requires. 

 Following the plan which we nave laid down in other articles, we shall 

 here describe the most important result* connected with the term in 

 question, referring for information on other matters to the following 

 articles: IXTIUHATIUN, FINITE ; GEXEHATI.NG FUNCTIONS, TIILOKY or ; 

 , X-ji-kTiu.v or ; OreuATlox ; EO.UATIO.NS or Dm 



We now see that, irregular as the first series may appear, the sue- 

 ive differences of the successive differences of its terms are always 

 the same, and we may thus extend the series further. Thus we 

 have 



83 97 118 131 151 173, 4c. 

 12 14 16 18 20 22, &c. 

 2 2 '2 2 2 2, &c. 



But still this question remains, assuming the preceding law of con- 

 tinuation, and also that we have thus the values of some function of .c 

 answering to x=l (namely, 43), -c = 2 (47), &c. the value of 



the function answering to fractional values of x ; for instance, when 

 x=24 .' Considering that x='2 gives 97, and j- = 3 gives 113, it might 

 seem at first sight that x = 24 should give 105. But a moment's 

 consideration will show that this can only be in the series 

 81 97 113 1-29, &c., 



and that the irregularity of the progression from term to term will 

 require a law to express it, such as will not allow of uniform pro- 

 gression between the terms. Such are the notions which might be 

 made to suggest themselves, and the difficulties of which find their 

 answers in the mathematical consideration of the subject. 



Let any term chosen at pleasure in a series be called a, let the 



i next term be a,, the next a,, and so on : that is, a, means the nth 



term from a, not reckoning a. The succession of Jirst diferciica (a 



more convenient way of expressing the Jint tucceteian of differences) 



is 



, , j if J . -:, 



The succession of second differences (tecoitd tuccession of differences) is 



j , (n l a), or a, 2a l + it 

 Oj a, (a, oj, or a, 20, + a,, &c. 



The succession of third differences, similarly derived from the pre- 

 ceding, is 



From which it may be made evident to any one who kn. 

 binomial theorem and the law of its co-efficients, that the first term of 

 the nth succession of differences is 



. 



(n + 1 t 



It we c .1 ry the hand along a sheet of paper, laying down points with 

 a pen at various intervals, we get a number of pomU, through which 

 an infinite number of curves nuy be made to pass; but, generally 

 peaking, there i* one which is more simple than all the rest. If we 

 also uugn various number*, we may conceive them all to be value* of 

 some fun-: lable jc, answering t.> Jt\, x 2, tc. Thus we 



u : what i that function of jc of which 



It is usual to denote this by A', the letter A standing for the ope- 

 ration of taking the difference, the exponent expressing th;it this 

 operation has been repeated until it has been performed n ' 

 and a being the term of the series used in the first operation. The 

 symbol is called the nth difference of a. 



I f , then, we write the series and its successions of different'' 

 using the results of the operation, but tlu-ii 

 follows : 



A 3 



A 2 



Ac. 



From which we find that a t =-a + A. 



1 = a + Aa + (Aa-(- A'o) 



, = a + A a 



+ 2 (Aa + A'u) 



or, a, = a + 3Aa + 3a A'a-HA'a. 



Proceeding in this manner, and by the assistance of the binomial 

 theorem as before, we find that 



^^ A'a+ . . . . (n + l terms). 

 two theorems are the fundamental pails of tho v.-hole theory 



