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 



P = Po{\ + f ) 



M = I'o (I + 0-) 

 T = T- {I + t) 



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

 neglect. 

 From the following equations, 



Jt =-^^^^z + d' ) (3) 





which take the place of (1), we eliminate u, ic, a, by differentiating the 

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

 ing to t. 



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

 be considered as a given lunction ot x, z, and t, but e as a lunctiou of 

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



dx-" dz' BT,;)z ETo X' I 



Bl'o ds RtXrTo To dz J RTo ,M' ] 



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

 consider the simplest case of linear vibrations. 



II. LINEAR VARIATIONS. 



When </ = 0, and t and e depend only on / and x, equation (A) be- 

 comes 



^-RTo^^ = ''^ (4a) 



^t^ ^x- df^ 



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

 vibrations in the atmosphere which gives c — ^ 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,dT + pdf^\ = 0,. To dr ~ RT„ da = C„ To dr - RT^ ds 

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



