Suppose A is an integral operator on the interval with kernel :We write to specify that the kernel of A is . Denote by K an integral operator of this type with kernel of the form and write Also let where k=0,1 and j=1,2.
The coupled equations which imply (2.7) can now be stated.
Consider the kernel (2.5) with on so that in the setting of Definition 2.1 and . With the notation , , we havewhere the dashes denote differentiation with respect to t. The theory of Tracy and Widom  allows equations for the quantities of Definition 2.1 to be derived which imply the equations of Proposition 2.1. Before presenting these equations let us show how (2.6) and (2.7) can be derived from the equations of Proposition 2.1.
First consider (2.7). We multiply (ii) by p, multiply (iii) by q, add and use (v) to obtainand consequently Substituting (3.2) and (iv) in (i) gives which relates tR to pq. On the other hand, another equation relating these two quantities is obtained by squaring (iv) and the first equality in (2.12) and subtracting: Solving (2.14) for pq (it follows from a small-t expansion that the negative square root is to be taken) and , substituting in (2.15) and introducing the notation gives (2.7). The boundary condition (2.8) follows from the fact that as and the corresponding behaviour of deduced from (7).
To derive (2.6) we simply substitute (2.16) in (vi) and integrate. The factor in the upper terminal of (2.6) results from changing the mean eigenvalue spacing from to .