176-178] Equations of Equilibrium 157 



The conditions for equilibrium to be possible are, therefore, three of the type 



In general, the result of eliminating the unknown variable p from these 

 equations is* 



HI 1 j. H i i .I. "7 / i r /Q | 7O\ 



a -o *~ I T *1 I 5 o~i T I o 5 =0 (^72). 



\ 3^ dy/ \9# Bf / V 9y & / 



If, however, the gas is at the same temperature throughout, the inter- 

 molecular pressure nr will depend solety on p, so that we can regard P as a 

 function of p only, and equation (371) can be written in the form 



= - 



There are of course two similar equations for H, Z, and the system of 

 three equations taken together simply expresses that the forces must have 

 a potential. This condition being satisfied, that expressed by equation (372) 

 is satisfied also, but the two conditions are not identical, It is only in the 

 case of a gas being at the same temperature throughout that the one can be 

 deduced from the other. 



If we suppose the potential to be ty, the equations of equilibrium are 



_<ty = lc)P 

 dx p dec 

 and two similar equations, and these have the common integral 



-/r=|- - (374). 



Now P = is + ^r by equation (370). If we ignore intermolecular forces 





and put nr = 0, we have P = ^y- = ^ , and equation (374) becomes 



or p = 



In putting isr = 0, we not only suppose the field of intermolecular force 

 outside the molecules to be very small, but also that the pressures across 

 any surface set up by collisions between molecules are very small. In other 

 words we assume that the size of the molecules is negligible. Thus the 

 equation obtained agrees, as it ought, with the equation 



p = De-**, 



obtained in Chapter V. (equation (165)), for the % of that equation had the 

 same meaning as the present m-^r. 



* Besant and Eamsey, Hydromechanics, i. p. 12 (1904). 



