I am currently writing a series titled "Sizing Conductors" in Electrical Contractor Magazine. Find the conductor size in this example before it is published in the September issue.

"What size THHN copper conductors are required to supply a branch circuit under the following conditions? The load will be a 39-ampere, continuous load. These branch-circuit conductors will be in a raceway. There will be a total of eight current-carrying conductors and an equipment grounding conductor in this raceway. The terminations on both ends are rated at least 75°C. The maximum ambient temperature will be 40°C."

here is my shot at this.

39A x 125% = 48.75A load continuous duty

in the 75 deg C column for termination design
a 48.75A load = #8awg minimum wire size allowed

next apply adjustment factors
48.75A / 0.70 = 69.64A for 7-9 current carrying conductors

69.64A / 0.91 = 76.53A for 40 deg C ambient temperature for THHN

minimum THHN from 90 deg C column for 76.53A load = #4awg

I'm not an engineer just self taught, so this may not be correct

My guess is #6....

http://www.electrician2.com/calculators/wireocpd_ver_1.html

What is the voltage and how far is the load from the source for possible voltage drop calculations?

Just to clarify so that no looks for 90 Deg C equipment terminations. Your comment "rated at least 75 deg C", to the best of my knowledge there is no equipment rated greater than 75 deg C, terminations yes, equipment no. It is even worst on UL508 listings, most control is 60 Deg C.

I have seen equipment with 90deg lugs but the equipment itself is only 75 deg rated

The fact that the current is a continuous 39 amperes is relevant to selecting the overcurrent protection device, not the conductors. The derating for eight current-carrying conductors is 70%. 39 amperes /70% = 55.7 amperes. For 75 degree C terminals, #6 has the required ampacity (65 amperes).

Regarding 90 degrees C versus 75 degrees C- in a pinch I've sometimes used the 90 degrees C rating for conductors for runs between splice boxes, where the splices themselves are rated 90 degrees C. Conductors at a temperature of 194 degrees F is not a safe situation for anyone working with them, though!

Have I been misunderstanding NEC 210.19(A) all this time?