80 Mr. J. Rose-Innes on the 



The next step was to find a formula for b. Now 



7 R a 

 6=- 4--. 



V T 



and the conditions already discovered with respect to the 

 algebraic expression for t considerably restrict the field of 

 research; this is an advantage, as it lessens the amount of 

 arithmetical work to be performed. I found that fairly oood 

 results could be obtained by putting 



&=-( 1 + 



v \ 



v + k — 9 -, 



v 



where R and k have the values already given and 



<?=7-473, # = 6-2318. 



By combining the formulfe given above for a and for b, we 

 obtain as the formula for the isothermals 



RTi' e \ I 



»= — I 1 + 



J) 



v \ g ) v + k' 



v + k— % 



where R, e, k, g, and / are constants, and have the values 

 already given. 



In order to test this formula it is desirable to draw a system 

 of isothermals, but if this be done in the ordinary Andrews' 

 diagram the result is not satisfactory, as the range of p is so 

 large. It was found possible to obtain a good diagram, how- 

 ever, by calculating pv and plotting it against v~i ; the calcu- 

 lated isothermals are shown as continuous lines, while the 

 experimental values are put in as dots. It will be seen that 

 there is a fair agreement between calculation and experiment 

 down to about vol. 3"4. Below this volume there is no longer 

 any agreement ; we should naturally expect such a result, 

 since the formula for a admittedly holds up to the neighbour- 

 hood of vol. 3 - 4 only. Therefore, even if the calculated 

 isothermal were to agree with the found isothermal below 

 vol. 3'4 for some one temperature, this would only happen by 

 a compensation of errors, and could not occur for any second 

 temperature. 



It will be noticed that in the neighbourhood of vol. 16 

 there is a sensible divergence between the calculated and 

 found isothermals amounting to slightly over 1 per cent. 

 This divergence is certainly unsatisfactory as far as it goes ; 



