analysis - Where does Jacobi's accessory equation come from?

I'm reading Charles Fox's An Introduction to the Calculus of Variations and in section 2.4 he just suddenly introduces Jacobi's accessory equation and I don't understand where it's coming from.

Jacobi's accessory equation (which oddly doesn't have a Wikipedia page) is $$\left[\frac{\partial^2 F(x,s,s')}{\partial s^2}-\frac d{dx}\frac{\partial^2 F(x,s,s')}{\partial s\partial s'}\right]u-\frac d{dx}\left(\frac{\partial^2 F(x,s,s')}{\partial s'^2}\frac {du}{dx}\right)=0$$ and has something to do with the second variation of $\int_a^b F(x,y,y')dx$ evaluated near a stationary path $y=s(x)$.

Could someone either derive (or give the main idea of the derivation) the equation or suggest a good source that does?

P.S. I checked in Gelfand and Fomin's book which I'm not reading but have laying around. The problem with their derivation is that it's in the middle of their book (as opposed to near the beginning of Fox's) and thus seems to make use of stuff I haven't gotten to yet in Fox's book.

---

Here's the context in Fox's book:

Note that for brevity the notation $F_{00} := \frac{\partial^2 F}{\partial s^2}$, $F_{01} := \frac{\partial^2 F}{\partial s\partial s'}$, $F_{11} := \frac{\partial^2 F}{\partial s'^2}$ is used in the following.

We just started looking at the second variation and trying to derive the conditions for extremizing the functional subject to weak variation. In the previous section we derived that if $t(a)=t(b)=0$, then $$\int_a^b \left(t^2F_{00} +2tt'F_{01}+t'^2F_{11}\right)dx = \int_a^b\left[t^2F_{00}-t^2\frac d{dx}(F_{01})-t\frac d{dx}(t'F_{11})\right]dx$$

Then this section starts off with:

On solving the [Euler-Lagrange equation], the equation of the extremal $y=s(x)$ which passes through the given points $A$ and $B$ can be determined.

Thus the quantities $F_{00}$, $F_{01}$, $F_{11}$, and $\frac d{dx} F_{01}$ can all be expressed in terms of $x$ and the differential equation $$\left[F_{00}-\frac d{dx}F_{01}\right]u-\frac d{dx}\left(F_{11}\frac {du}{dx}\right)=0 \tag{1}$$

can then be solved for $u$ as a function of $x$. This is an ordinary linear differential equation of the second order and is known as the subsidiary or Jacobi's equation or, more frequently, as the accessory equation.

On taking $x$ to be independent and $t(=t(x))$ the dependent variable in the integral $I_2 [= \int_a^b \left(t^2F_{00} +2tt'F_{01}+t'^2F_{11}\right)dx]$, it is easily seen that $(1)$ is the [Euler-Lagrange equation] for minimizing $I_2$ with $t$ replaced by $u$.

Note that other than that last line he doesn't explain what the function $u$ is.