112 Prof. W. E. Story on Partial 



From (41) and (51) follows 



".-.-^M/-^ ...,,«...,"<■<•..&■ KM) 



(r=l,2,3,...,K-l), 



where, by (57) and (58), the constant term of the right 

 member is 



fl »- a °- In P. + 2 2 (,_!)• ( G-1) c • W 



Also, by (28) and (33), 



— "«= i »(;;s) («) 



Equations (59) and (01) serve to determine &> and ulti- 

 mately u T 0'=1, 2, 3, . . . , #— 1) and the formulae (28) for 

 the partial pressures in mixtures of any given components 

 from actual observations of mixtures of those components in 

 different proportions. Such an observation is supposed to 

 give the values of m v « 2 , x s , . . . , w K , p l : p K , p 2 : p K , p s :p Ky 



• • • > Pk-i : P K - From these values are determined 

 u-u K (r = 1, 2, 3, . . . , *-l) by (61) and z k (k=l, 2, 3, ... , 

 /c— 1) by (33). On substituting the value of u r —u K for any 



value of r and the values of the z's in (59) we obtain a linear 

 equation in the coefficients e„ for 2 ^ G and in the con- 

 stant a^—a^; each observation gives k— 1 such equations 



in the c's and in the k— 1 constants, corresponding to the 

 /c— 1 values of r. So far as we know, w is an infinite series 

 in the z'e, but in practice we must suppose that it is con- 

 vergent and that we shall get a sufficiently close approxi- 

 mation to it by taking a certain number of terms of it. 

 As there will generally be no reason to assume that there 

 is any difference in the order of magnitude of the different 

 #'s and, therefore, of the different z's, the terms of w that are 

 of one magnitude are those that are of one degree in the z's, 

 that is those lor which G, in the notation of (47), has one 

 value. The natural mode of procedure will, then, be to take 

 the aggregate of terms of to whose degree does not exceed 

 a certain number as a sufficiently close approximation to the 

 whole expression. In other words, we agree to neglect every 



