154 UNDULATORY THEORY OF LIGHT. 



to A'B. Let the natural elasticity of the air be cnllcd c. By this natural 

 elasticity we mean that force with Avliich the air would tend t-o expand into a 

 vacuum, or crush in the sides of a vessel from the interior of which the air has 

 been removed. It is measured by the weight upon a square inch, of the column 

 of mercury required to balance it, and tiiis is compared with g-ravity by con- 

 sidering that it would move a body of a unit in bulk and density (say a cubic 

 inch of water) as much faster than gravity alone, as it is greater than the Aveight 

 of such a unit. If, then, g represent the velocity which gravity can impart in 

 a second, if h be put for the height of the barometric column, and if D stand 

 for the density of the mercury in the column, 



e—gJiD. 



Emidoyiug e' to denote the increased elasticity of the compressed air between 

 AT' and BE, and putting AB=;a;, A'Bizz.t', and AA'z:=a;", we have 



e' = e— : e — 



X 



This difference, e' — e, is the effective force by which BE is acted upon ; for 

 the space between BE and CD is filled with uncompressed air, which opposes 

 the movement of BE by the force e, while the compressed air between A'E' 

 and BE urges it with the force c. Now, the velocity of movement imparted to 

 a body by a given force i'^ dejoendent not on the time and force only, but also 

 on the mass to be moved. And it will evidently make no difft^rence in the 

 rapidity of propagation of movement, whethei- — mass remaining tlii- same — we 

 suppose BE to be the stratum of molecules which is nearest to AF, or whether 

 we suppose it to be the middle stratum of a number filling the space half way 

 between it and AF on the one hand, and CD on the other. In either case 

 there will always be the same mass [m) which will be proportional to the length 

 of the column x, and its density, which we may represent by d : — or vi will 

 always be as xxd. Now the velocity v. which any force (as e' — e) will gene- 

 rate in the miuute time t in this mass, will be 



e'-e - 



m 



or, substituting the values of e' — e and ?n, 



ex"t 



v= . 



. XX a 



And the velocity of wave-propagation (which may be denoted by V) multi- 

 plied into the time t, will give x, the distance the tremor has advanced in that 

 time; or x^iNt. Moreover, since x" is the distance moved by the molecules 

 in the time t, it must have the same ratio to x, or to x', (for the di.fference is so 

 slight as not sensibly to affect the ratio) that v has to V. 



XT ^' ^" 



Hence, — = — . 



V x' 



Substituting these values, we have 



''•==/v'r '"■•^"=^^""'^^'-^',2 



-V^- [^- 



But c and d are constants. Ilence the velocity of the wave is uuiioim. ' 

 In this demonstration it has been tacitly assumed that the expansive force of 

 a confined body of elastic fluid is inversely ])roportional to its bulk. This is 

 called the law of Mariotte; and it is true if the temperature of the finid, when 



