Steam of Maximum Density. 177 



com 



1™t> / *fi n + l f , 7l + 1 rc + 2* 2 n + 1 « + 2 rc + 3* 3 , Q i 

 logP,= ^(l--_ -+_._-,__ ^ i-^a + ^c.} 



= to{l-l-151292- + l'267414- 2 -l-36329o4 + &c.T. 



The above series, representing the true law of elastic force of 

 steam, may be used to show in what manner Roche's law and 

 two other approximate laws may be formed. According to the 

 true law, and also according to the three approximate laws, the 

 common multiplier of all the terms of the four different series is 

 kett — the quantity a being the hyperbolic logarithm represent- 

 ing the rate of increase per degree when the thermometric tem- 

 perature is or dt. Also the true law as well as the three ap- 

 proximate laws agree in having their two first terms the same, 



A«nl ^ ). It is at the third term only that the varia- 

 tions begin. 



The three approximate formulae are these : — 



71+1 



logP f =*a^l + ^ 2 (No. 1), 



logP,=fo«f(l + ^±1 ^'(No. 2, or Roche's formula), 



n+l t 



\og~P t =k*te 2 a (No. 3). 



Putting — — or ry =m, we shall have for the three approxi- 

 mate formulae for log Y t > 



1 + -) , katll + m-J , and kute~ m «. 



The coefficients of the third and fourth terms of the above three 

 approximate formulae are, omitting the common multiplier ku, 

 as follows : — 



+ 1-238383-1-300835 (No. 1), 



+ 1-325473-1-526007 (No. 2), 



+ -662736- -252668 (No. 3). 



The coefficient of the third term of the true formula being 

 1-267414, it will be seen that the coefficient of the same term 

 in Roche's formula is -058059 in excess, and in formula No. 1 

 is -029031 in defect. That is, Roche's approximate formula 



