Question about the meaning of an important footnote to entry 109 in Gauss's diary Entry 109 in Gauss's diary (and the related material in Gauss's Nachlass) is the main subject of David Cox's famous article "The Arithmetic-Geometric mean of Gauss". This entry deals with the agm of two complex numbers $a,b$, taken as a multi-valued function. It reads as follows:

Between two given numbers there are always infinitely many means both arithmetic-geometric and harmonic-geometric, the observation of whose mutual connection has been a source of happiness for us.

The harmonic-geometric mean of two numbers $a,b$ (which we denote as $H(a,b)$) is $M(a^{-1},b^{-1})^{-1}$. According to Cox, this entry is a continuation of a previous study of the agm, which resulted in the following relation between the infinitely many values of the agm:
$$(1) \frac{1}{(\mu)} = \frac{1}{M(a,b)}+\frac{4ik}{M(a,\sqrt{a^2-b^2})}$$.
Relation (1) is not the most general formula for the different values of the agm - this is written as:
$(2) \frac{1}{(\mu)} = \frac{d}{M(a,b)}+\frac{ic}{M(a,\sqrt{a^2-b^2})}$, where d and c are relatively prime numbers and $d \equiv 1\pmod{{4}}, c\equiv 0\pmod{{4}}$. The main purpose of part 3 of Cox's article is to check if Gauss only knew (1) or that he even knew (2) (Gauss never stated (2)). Looking into diary entry 109 in volume 10-1, i saw an additional piece of evidence that is not considered at Cox's article; this is a footnote that Gauss writes next to this entry, and reads in Latin as:

Terminus constans expressionis
$$\frac{Ad\phi}{\sqrt{f+2gcos\phi + hcos^2\phi}}$$
est Medium Geometrico harmonicum inter
$$\frac{A}{\sqrt{\frac{\sqrt{(f+h)^2-4g^2} + f-h}{2}}}$$ et $$\frac{A}{((f+h)^2-4g^2)^{\frac{1}{4}}}$$.

Therefore my questions are:

*

*According to Google translate, the Latin phrase "Terminus constans expressionis" means "The border consisting of expression". The use of this phrase in this context is therefore unclear to me, so I'd like to know it's meaning.

*The first expression in Gauss's footnote seems to be a differential expression (it involves $d\phi$). Therefore, Gauss's statement is probably about the equality of an integral (between which limits?) of the first expression and the Harmonic-geometric mean of the two last expressions. The question remains: what are the limits of integration (from $\phi = ?$ to $\phi = ?$)?

*Schlesinger remarks that the expression "has been a source of happiness for us" was a kind of code-expression that Gauss used to refer to new results in number theory, and speculates that the appearence of this expression in diary note 109 means that Gauss understood the connection between this note and the theory of binary quadratic forms. Also, the notation of Gauss in his footnote (he uses $f,g,h$) reminds one of his notation for quadratic forms. Therefore, what is the relation of this footnote to binary quadratic forms?

 A: I finally dechipered what was Gauss's intention in his Latin footnote, but i still don't have any idea how he arrived at this result.
The Latin phrase "Terminus constans expressionis" seems to mean the "constant term" $A$ in the Fourier expansion of $\frac{1}{\sqrt{f+2gcos\phi+hcos^2\phi}}$, (the trigonometric series $A+A'cos\phi+A''cos2\phi+...$) which is actually equal to $\frac{1}{\pi}\int_0^{\pi}\frac{d\phi}{\sqrt{f+2gcos\phi+hcos^2\phi}}$. This interpratation of Gauss's Latin is consistent with Gauss's notation in other places in his writings - for example, in p.52 of Cox's article, Cox mentions Gauss's result that $\frac{2}{\pi}\int_0^{\pi/2}\frac{d\phi}{\sqrt{1+\mu cos^2\phi}} = M(\sqrt{1+\mu},1)^{-1}$ - a result that Gauss describes by using the same Latin expression (the meaning of this result is that the constant term in the Fourier expansion of $\frac{1}{\sqrt{1+\mu cos^2\phi}}$ is $M(\sqrt{1+\mu},1)^{-1}$).
Note also that the Gaussian result that is mentioned in Cox's article is a very special case of Gauss's much more general result stated in his Latin footnote; put $f = 1, g =0 , h = \mu$, and you will get this special case. Also, In the special case that $h = 1, f = g^2$, than we have to calculate a mean of two equal numbers, and therefore the "constant term" $A$ is $\frac{1}{\sqrt{g^2-1}}$, a prediction that was verified by me by taking specific numerical examples: for example, if $h = 1, g =2, f=4$, than wolfram alpha says the result of the integral is $\frac{1}{\sqrt{3}}$. In the latter case the indefinite integral is elementary, and it's also clear from the fact that we take the hgm of two equal numbers (there is no need to activate the hgm algorithm), so this somehow hides the true theory behind Gauss's statement.
Regarding the numerical verification of Gauss's result in cases where there is a need to run the hgm algorithm, i verified Gauss's result using online agm calculator and wolfram alpha integrators. The true significance becomes clear in such cases; for example, if we take $f = 24, g = 5, h = 1$, than the result of both the integral and the hgm is $0.208811$.
So this was Gauss's statement, but i still want to know if this statement is known to the mathematical community, and understand a bit more about how Gauss arrived it.
