870 Mr. S. D. Wicksell on 



We then have 



and putting 





R = 



1 





#0 



= «, 





1/v 



= 47, 



P 



0o 

 ' 6 



=y> 



we get 



y(l-&.p)=0— aa* + a&^ (2) 



I now want x expressed as a series in powers of y or, which 

 is the same, in powers of P. 



The first question is how and when this series is con- 

 vergent. 



The equation (2) gives 



*=F(y), 



and the series is convergent for all values of y less than the 

 absolute value of the least y that gives F(y) a singularity. 



It is easily seen that F(?/) can have no other finite singu- 

 larity than when it ramifies, and that occurs when y has such 

 a value that the cubic (2) has equal roots. For every value 

 of the parameters a and b, that is of the temperature, there 

 are three such values of y, always one negative and two 

 positive or imaginary. They are given by the equation 



My +y^l2b* + 8*b 2 ) -y(20*h-12b 2 -±* 2 ) +Ab-u = 0^ 



It is seen from this equation that when a = 4& or 



6= -j0 o one of the roots is equal to zero. The least absolute 



value of the roots of the equation is the interval of convergence. 



For 6 — jr 6 this interval must be equal to zero. It is also 



easy to see that when 6 grows the equation has two positive 



roots and one negative, which always grow until 0= ^=- O , 



when the positive roots coincide and then turn imaginary. 



At 0=£^t#o the interval of convergence is |yl = ^r. 

 Z 4 b ob 



6= ^fj6 is, theoretically from van der Waals's equation, 



