DIFFERENTIATION 



I H;. \.M.M A 



811 



Between the <'alculus of Finite Difference* and 



tin- Differential <'alculus (see CAI.I i 1.1 i) ia title 

 which means tlie calculus of inHnitc-jmal differ- 

 ences) there an- m.-uiy ini|Hi taut points .| run 

 ti.-i-t aiid of -imilarit\, which would \- not le-s 

 clearly appreciated if the. names were changed, a* 

 I'.onle all l)iit suggested, the former to Calcula- of 

 Differences, tin- latter to Calculus of Limits. 



Tin* methods uf the Calculus of Differences are 

 in vogue among actuaries and others in dealing 

 with MatiMics such as mortality table* ; and from 

 this calculus are derived many formulae of approxi- 

 mation of great practical value, such as the rules 

 for finding the area of surfaces bounded by curved 



Hues. 



Differentiation, that organic progress which 

 occurs when certain parts of a uniform whole 

 heroine structurally different from the others, or 

 when, in other words, the homogeneous becomes 

 heterogeneous. Inequality in internal and external 

 conditions of life brings about restriction of certain 

 \ it'll processes and the predominance of others, 

 and as this division of function is established, 

 diversity of structure results. Differentiation i- 

 the structural change which is associated with the 

 [>h \siological 'division of labour,' and the process 

 is essentially the same whether it find expression 

 in cells, tissues,' organs, or entire organisms. See 

 DIVISION or LABOUR, EVOLUTION, VARIATION, &c. 



Diffraction. In general, light is propagated in 

 straight lines in a homogeneous medium : but, if it 

 be caused to pass through an opening which is not 

 large in comparison with, the wave-length of the 

 light, the law no longer holds. Such phenomena 

 are said to be due to diffraction. The subject will 

 be found treated at greater length under LIGHT. 



Diffusion. The particles of all material 

 bodies, except such as may be totally devoid of 

 heat, are in rapid motion. In the case of solid 

 bodies the excursions of any one particle are 

 limited to a small space ; but in fluids a particle 

 may move more or less freely throughout the whole 

 space occupied. This intermixture of molecules 

 may occur also when different fluids are placed in 

 contact with each other, but it may be prevented 

 by the existence of tension at the common surface 

 (see SURFACE-TENSION and CAPILLARITY). When 

 it does occur, the fluids are said to diffuse into each 

 other. 



Diffusion of Liquids. The diffusion of dissolved 

 salts may obviously be considered under this 

 heading. The phenomenon may be conveniently 

 studied by introducing a strong solution of some 

 highly -coloured salt, such as bichromate of pota-h. 

 into the bottom of a tall glass cylinder nearly 

 filled with water. The rate of diffusion varies 

 with the nature of the liquids. Graham was the 

 fir>t to investigate the subject carefully. Me tilled 

 a number of similar glass vessels with solutions of 

 different salt*. The mouths of these vessels were 

 carefully ground so that they could ! closed by 

 means of glass plates. The different vessels were 

 tlien placed in equal glass jars, and covered with 

 water to a definite extent. Next the j,'lass covers 

 were cautiously withdrawn, and the diffusion was 

 allowed to go on for a certain time. The rate at 

 which each liquid diffused was thus obtained. 

 (iraham found that, for anv one solution, the rate 

 is proportional to the gradient of concentration 

 i.e. to the rate at wnich the quantity of salt 

 dissolved per unit- volume varies per unit length. 

 Thus the law regulating diffusion of liquids U 

 analogous to that which regulates the conduction 

 of heat in a homogeneous solid. Hence the equa- 

 tions obtained by Fourier in his Thtorie fie la 

 Chaleur apply to the problem under consideration. 



Graham found also that rise of temperature 



r.. ... , ||. ... 



ii to two CIAMM*, Coltoidi and CVyto//W< 

 mem!..-!- of the firNt chum diffusing very much 

 more -lowly than tin me of the Ncond. I 

 1 1-. in. .n~ have been much extended by m<>n> rcioenl 

 observers employing varioun method* of observa- 

 tion. 



If two mi-cible liquid* be eparated by a mem* 

 brane of bladder or of |>arrhmcnt ]: 

 d illusion takes place through the M-ptum at mt- 

 which are usually very different for dilS 

 liquids. This phenomenon i* known a* Otmott 

 (q.v. ). It was lu-t shown by Nullet that. 

 M ! filled with alcohol V dOMd by 

 bladder and placed in water, the dtfwIoM of th- 

 water is so much more rapid than that 

 alcohol that the bladder is burnt becntiM* of th>- 

 increase of the contents of the ve*>*l *lu> 

 closes. By this means the various n-titu-nu of 

 a mixture of colloid and crystalloid sulwtance* 

 may lie separated to any de-ned extent. '1 h. rate 

 at which liquids diffuse into each other through 

 a septum depends greatly upon the molecular action 

 between them and the seiitum. 



Diffusion of Gases. If two flasks, each filled 

 with a different gas at a given preiwure and 

 temperature, be placed in communication with 

 each other, the gases will lie found to intcrdiffuae. 

 The rate of interdiffusion is shown \>r tin-.' 

 be nearly in inverse projiortion to the square root 

 of the product of the densities of the two gaM0, 

 and the experimental results are in accordance 

 with the t henry . 



Effusion of" Gases. This is exhibited in th 

 passage of a gas into vacuum under constant 

 pressure through a small opening in a verv thin 

 plate otherwise imjerviou8 to it. The wort done 

 in the passage of a given volume of the gas it 

 proportional to the pressure, and the equivalent 

 kinetic energy is proportional to the product of 

 the density and the square of the speed of effusion. 

 Hence the speed for a given pressure varie* in- 

 versely as the square root of the density. Graham 

 showed that this result of theory is closely realbed 

 by experiment. He showed, further, that when a 

 discrepancy exists, it is due to the finite thicknem 

 of the plate. . 



Triinxjiinition of gasm is the term to the peMBji) 

 of gases under pressure through a tine capillary 

 tube. This subject was also investigated by 

 Graham, who found that the rate of pjuaage M 

 not affected by the material of the tube. Thi 

 seems to indicate that the till* become* coated 

 internally with a thin film of gap, 00 that the 

 op|Mihition to the flow of gas is due to Yi*r 

 (q.v.). 



The rates at which different gates pan through 

 fine unglazed earthenware are inversely a* the 

 square roots of their densities. Hence we hare a 

 means of separating gases the densities of which 

 are different (see AlMoi.YSIS). If the -p'.tim I* 

 made of caoutchouc, which ix not |>mu-. the pas- 

 sage of gases still O4i-urs. The gas *eenu to com- 

 bine with the matter of the neptum on the oneMde, 

 to diffuse through it. and finally to U given off on 

 t he ot her side. The paMage of MOM ee, Mich a* 

 carlMinic oxide, through hot cant-iron IK analogotta. 



Diffanima i* the name given by the gram- 

 marians of the 1st century to .. the nlxth letter 

 of the primitive (Jrtfk alphabet, which had U- 

 come obsolete, and was only known to them from 

 inscription*. The name was given owing to a 

 fancied resemblance of it* form / to a doable 

 fftintma. Its sound was mnnething like that of 

 our IT. It b found in IVhqMtnneiMan in-cription 

 as late as the 6th century B.C.. but it had dis- 

 appeared from the alphabet of Attica before the 

 date of the oldest inscription* Le. before the 



