360 Mr. Rankine on the Hypothesis of Molecular Vortices, 



The lowei" limit of integration of ^ must be made — co , that 

 it may inchidc orbits of indetiuitely small magnitude described 

 round the atomic centre. 



The nature of the function -^ is limited by the following con- 

 dition : — 





Let 



2Q 



^/^ (^) 



M+l = ' 



Then these transformations give the following result for the 

 pressure at the bounding surface of an atom : — 



V 



hfj, _ W'^i e/c(^-0,) ft) ,, 



hjjb 6 CO I 



CO, 2- + -32 - -m- + &C- 



^a 



(8) 



(o\, &c. being the successive differential coefficients of co with 

 respect to kef), when cj) = cf)^. 



(4.) The following transformation will be found useful in the 

 sequel. 



Let X be the indefinite value of log^V, and X, its actual value 

 in the case under consideration. Let Gr be the same function of 

 X which CO is of k^, and let G', G", &c. be its successive differ- 

 ential coefficients with respect to \. 



Let 



Then 



hflGy 



^~MVH, ^^^ 



The function H has the following properties, which will be 

 afterwards referred to : — 



1^ + ^H.-G,=0l 



^^' I (10) 



(5.) Case of a Perfect Gas. — As a substance is rarefied, it 

 gradually approaches a condition in which the pressure, under 

 like circumstances as to heat, varies proportionally to the den- 

 sity. This is because the effect of the molecular attractions and 

 repulsions on the pressure diminishes with the density, so that 



