388 Prof. R. Clausius on the Theoretic Determination of 

 namely 



N = ^( 1+ 3 1 5 V+ 3TFr7 X4 + -) ; ' 

 and then put 



«, = M-N\, (21) 



W=M+NX (22) 



From the last two equations it follows that 



W + w=2M, . . ' (23) 



Ww=M 2 -N 2 \ 2 , (24) 



from which it is evident that the sum and the product of the 

 two quantities TV and w contain only even powers of A,. Now, 

 as in the expressions of II and 6 given under (8) and (16) 

 the quantities "W and id occur only in combination as sum and 

 product, it thence follows that the quantities II and 6 also 

 contain only even powers of X. This implies that in the vici- 

 nity of the critical temperature, where X approaches the value 

 nil, the quantities II and 6 behave in an essentially different 

 manner from the quantities TV and to. We shall revert to this 

 further on. 



§ 5. The equations hitherto evolved, presenting the four 

 quantities w, TV, II, and 6 as functions of X, of course thereby 

 determine indirectly the connexion in which each of the three 

 quantities w, TV, and II stands with the quantity 6; but this 

 indirect determination, effected by means of a third quantity, 

 does not completely satisfy the requirements. In investiga- 

 tions of the process of vaporization, the temperature is usually 

 considered as the given quantity, and it is wished to deduce 

 from it immediately the pressure of the vapour and the volumes 

 of the vapour and the liquid. In this sense, therefore, must 

 we render more perfect our mode of determination. As in 

 our developments hitherto the temperature does not occur 

 explicitly, but only the yet undetermined temperature-function 

 6, we must provisionally attach the determination to this tem- 

 perature-function and set ourselves the problem of arranging 

 so that from the value of 6 the corresponding values of w 9 TV, 

 and II shall be obtained in the simplest possible manner. This 

 I have endeavoured to accomplish by calculating a table giving 

 for the different values of 6, increasing one hundredth at a 

 time, the corresponding values of X. From this table X can 

 easily be determined, by interpolation, for any value whatever 

 of 0; and when X is known, w, TV, and II can be directly cal- 

 culated with the aid of the above formulas. 



