Bill, here's the story.

RMS and average are basic mathematical functions whose definitions you
can find in numerous references(*). I'll state them here.

The average value of any periodic function is the time integral over a
cycle of the instantaneous value of the function, divided by the period.

The RMS value of any periodic function is the square root of the average
(mean value) of the square of the function, where the average is defined
as above.

First, let's look at these two values for a sine wave with peak
amplitude of V. The instantaneous value (value at any time t) is V *
sin(wt) where w is omega = 2 * pi * frequency. The integral over a cycle
is zero (since the wave spends equal amounts of time at equal amplitudes
above and below zero), so the average value is zero. Some careless
references will give a non-zero value for the "average" of a sine wave,
but this is really the average of the absolute value (that is, the
full-wave rectified value) of the sine wave. The actual average value of
a sine wave with no DC offset is zero. (If it has a DC offset, the
average value is simply the value of the offset.)

The RMS value of the sine wave is the square root of the average of the
square of the original sine function, which is V^2 * sin^2(wt). If you
graph this, you see that it looks like a rectified sine wave -- it never
goes negative. If you go through with the math to get the average of
this squared function, you get the nice value of V^2 / 2 for the
average, hence V / sqrt(2) ~ 0.707 * V for the RMS.

Now let's apply that sine wave to a resistor and look at the power.

The *instantaneous* power, that is the power at any instant, dissipated
by the resistor is v * i = v^2 / R where v is the instantaneous value of
the voltage: v = V * sin(wt). So v^2 / R = V^2 * sin^2(wt) / R. Look
familiar? So what's the average power? Using the definition of average,
the average power is the integral over a period of the instantaneous
power, divided by the period. In other words, it's average value of V^2
* sin^2(wt) / R. Looking at what we did to get the RMS voltage above,
you can see that the average power is simply the square of the RMS
voltage, divided by R.

That's why the *average* power is the square of the *RMS* voltage
divided by R. It's important to realize that this holds true for any
periodic voltage function -- square wave, triangle wave, what have you.

You can use the basic definition of RMS to calculate an RMS value of
power from the instantaneous power, but it's not useful for anything. A
resistor dissipating 10 watts of average power gets exactly as hot if
that average power is supplied by DC, a sine wave, or any other
waveform. That's not true of the RMS power -- different waveforms
producing the same average power and causing the same amount of heat
will produce different RMS powers. So average power is a very useful
value, while RMS power is not.

The only thing that makes RMS voltage or current useful at all or
worthwhile calculating is its relationship to the useful quantity of
average power.

(*)You were asking for references -- you can find the definition of
average on p. 254 and RMS on p. 255 of Pearson and Maler, _Introductory
Circuit Analysis_, and average on p. 423 and RMS on p. 424 of Van
Valkenburg, _Network Analysis_. You'll also find an explanation in both
books similar to the one I just gave. These happen to be the two basic
circuit analysis texts I have on my shelf -- you should be able to find
the same explanation in just about any other circuits text.

Roy Lewallen, W7EL