To find the change in J, consider the four neighbors of an odd site. These are the only spins which interact with it. The multiplicative contribution to the partition function from the sum over the two values of the spin at the odd site is: e J ( N + − − N − − ) + e J ( N − − − − N + ) = 2 cosh ⁡ ⁡ ( J [ N + − − N − − ] )