398 Dr. R. D. Kleeman on Fundamental 



function of v, and X, x, and Y such functions as will reduce 

 each of the equations (3), (4), (5), and (6) to the same form 

 as equation (19) if substituted for U, u, and Z, respectively. 

 Now in order that these equations should reduce to the same 

 form when the values of U, u, and Z given by the above 

 equations are substituted, we must have 



+ *i =0, (20) 



= 0, (21) 



J V dv x h T 2 + J W h T 2 

 C(dW 2 \ gT f / dw 2 \ ST 



J^l^gT + J^^gT+ f\.fc=0, (22) 



and ■W 1 -W 2 + ^ 1 -^ 2 =0. (23) 



Subtracting equation (21) from equation (20) and substi- 

 tuting for (Wj— W 2 ) from equation (23), we obtain z x = z 2 . 

 Now this can only be in general the case if z is independent 

 of v, that is a constant, which may of course be equal to 

 zero. The same result can be deduced from equations (23) 

 and (22). Equation (23) may be written W 1 + w 1 = W 2 -\-w 2 . 

 It will now be seen that this equation cnn hold in general 

 only if the quantities in the equation are independent of v, 

 and thus functions of T only. It follows then that Wi = W 2 

 and w 1 = w 2 . Now Wi or W 2 is equal to zero. For U or 

 X + W is by definition the work done per molecule against 

 molecular attraction on separating the molecules an infinite 

 distance from one another when their initial volume is v. 

 Now this work could not contain a term that is a constant 

 or depends on the temperature only, and thus is independent 

 of the initial and final volume of the substance. Wi and W 2 

 are therefore in general each equal to zero, and w 1 and w 2 

 each a function of the temperature which we will denote by 

 yjr(T). From equations (20) and (21) it follows now that 

 Zi = ? and z 2 — 0, or z = in general. We have then U = X, 

 u=w + w, andZ = Y. Since equation (1) must reduce to the 

 same form as equation (19) when we put U = X, u = x, and 

 Z = Y, it follows that 



TT v m a p c , /T v \ m a p c /T v \ /rpx 



It should be noticed that not only need w not be a 



and y 



