The author is establishing an upper limit for the mutual inductance M.

The missing text is:
"The energy stored in the circuit cannot be negative because the circuit is passive."

Can somebody explain equation 13.34 where he completes the square. This has me scratching my head. He pulls iii2√L1L2 out of the blue. I can't see how this is completing the square.

 

Let a=i1√L1 b=i2√L2

In 13.33 we have 1/2(a²+b²) .To write this in (a+b)² [to get the coefficient of coupling relation] we need an extra 2ab.so we add and subtract 2ab.But since we have 1/2 multiplying (a-b)² it is enough to add 'ab' to cancel the extra -2ab we have introduced to make a²+b² --->(a-b)².

so a²-b² becomes (a-b)² and ab (i1√L1 * i2√L2) has been added to get the relation for coefficient of coupling.