the Gas-equation. 871 



the critical temperature. At = co the interval o£ conver- 

 gence will be ' y \ — -. 



The series in powers of P has an interval of convergence 



P - a ft 

 c ~~ 21b 2 ' 



where 8 = for 0= -r^-0 o ; and then grows with 6 and becomes 



equal to 1 for 6= ^y^o; thereafter S grows infinitely with 6. 



It will be seen from numerical examples that the series 

 generally converges very fast. 

 In order to find the series we put 



x = a l y + a 2 if + azif+ 



and substitute this in the equation (2). 

 We then get 



y = a\y + «2,r + (-hy z + &d + a- y r ° + 



+ b(a } y 2 + a 2 if + a 3 i/ + ff 4 ?/ 5 + ) 



- a [a x y + 2a { a 2 if + (« 2 2 + 2a 1 a 3 )/ + (2a Y a, + 2a 2 a 3 > 5 + . . .] 

 + K«iV + 3r/ 1 2 ^ 4 + 3(« 1 2 « 3 + « 2 2 a 1 )j/ 5 + ....]. 



As « and 6 are supposed independent of P, or which is the 

 same, of y, w T e can put the coefficients of the powers of ij 

 equal to zero. 



That gives us 



a 2 = « — &, 

 a 3 = 2«a 2 — fra 2 — ba, 

 a± = <x(a 2 + 2a 3 ) — 6a 3 — 36aa 2 , 

 a 5 = 2«(a 4 + a 2 a 3 ) — ba± — 3&a(a 3 + « 2 2 )' 

 Putting a—b = (j) and solving the system we get 



a 2 = (f>,\ 



a, = 2tf-b 2 3 



a 4 = 5<£ 3 -56 2 <£-6 3 , 



« 5 = 14(£ 4 -21<^ 2 -6^<M 2Z> 4 . 



