Cyclic rule, for partial derivatives

Various rules about partial derivatives are expressed using the general variables A, B, C, D,. . . instead of variables we know. It will be our job to apply these expressions to the state variables of interest. The two rules of particular interest are the chain rule and the cyclic rule for partial derivatives. 

This is the cyclic rule for partial derivatives. Notice that each term involves p, V, and T. This expression is independent of the equation of state. Knowing any two derivatives, one can use equation 1.25 to determine the third, no matter what the equation of state of the gaseous system is. 

Now we invoke the hint. The definitions for a, k, and the partial derivative (dpldT)y all use p, T, and V. The cyclic rule for partial derivatives relates the three possible independent partial derivatives of any three variables A, B, C ... 

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