PAPER BY MAX MARGULES. 301 



The motion is caused by small variations of temperature. Such vari- 

 ations will, as a rule, produce only slight variations of density and of 

 pressure. If we put 



V = Po{\ + 

 M = /'«. (1 + ff) 

 T = T<> (1 + t) 



then e, <r, r, are small numbers whose products and squares we shall 

 neglect. 

 From the following equations, 



dt-- K±0 Jx 



^ + dx + ^ ~ W V^T + To ^ y»- u 

 6 = c-fr 



which take the place of (1), we eliminate u, w, o", by differentiating the 

 first according to x, the second according to z. and the third accord- 

 ing to t. 



We thus obtain the following differential equation, in which r is to 

 be considered as a given function of x, z, and t, but fas a function of 

 x, z, and t that is still to be determined. 



}x 2 dz> RT jz RT dt % I (4) 



g_dt 9(9 .1 * T o\ r - *-ft| 

 " .BT d* RT \RT o T <te / .BTo ^ 2 3 



Before we treat the equation for motions in two dimensions we will 

 consider the simplest case of linear vibrations. 



II. LINEAR VARIATIONS. 



When g = 0, and r and s depend only on / and x, equation (4) be- 

 comes 



ll-RT l £ = d ^ C - (4«) 



dt 2 dx 2 dt 2 



and when t = 0, this becomes the Newtonian equation for acoustic 

 vibrations in the atmosphere which gives c = V RT as the velocity of 



propagation. 



If we consider— not the variation of temperature, but the flow of 

 heat as known, then we have to introduce the relation. 



dQ =C v dT + i^(-) = G * T ° dT - ET > dG = G > To dr " RTo d€ 

 where the change of kinetic energy is omitted, as being a quantity of 



