34 SMITHSONIAN MISCELLANEOUS COLLECTIONS VOL. 51 



and, as the condition that the mass of each particle of fluid shall 

 remain invariable, we have 



o = d(pu) + d(pv) + d{pw) + dp 

 dx by dz dt 



In these equations the variations with the time, when x, y, z re- 

 main constant, are expressed by the differential with regard to t. 

 Since now in our present case of the earth's atmosphere, we 

 assume that at every point the pressure, temperature, motion and, 

 therefore, the density are invariable as to time, therefore, we may 

 in equation (II) substitute 



dp 



and also in equations (I) 



du dv dw 



Furthermore, we have 



= 



dt dt dt 



p.d P(l + kz) p 



or 



P (1 + kz) d p 



where Pis that value of the pressure (or the reduced barometric 

 reading that measures the pressure) for which the atmospheric 

 air at the temperature o° C. is o times heavier than the mercury 

 of the barometric column and k is the coefficient of expansion of 

 the air for one unit of the thermometric scale that is used to measure 

 the temperature or r. 



Since z is assumed to be independent of the time it can be ex- 

 pressed as 



z = f {x, y, z) 



where / indicates a known function, which, as above stated, is now 

 approximately known; therefore we may also write 



p P Pk . . , - , , 



~ = - + — / (x, y, z) = F (x, y, z) 

 poo 



where F (x, y, z) again indicates a known function which for brevity 

 we indicate by F* 



