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Abel's addition formula for algebraic integrals is one of the most celebrated results of nineteenth century mathematics.

This is my attempt to describe, in the simplest possible terms, the essential idea's behind Abel's addition theorem, based on my reading of Skau[1], Houzel[2], Griffiths[3] and Roberts[4].

Then $x_i$ and $y_i$ are algebraic functions of $t$ and you can think of them as $n$ different algebraic parametrisations of the curve $F(x,y) = 0$.

By differentiating $F(x_i,y_i)=0$ and $G(x_i,y_i,t)=0$ with respect to $t$, it is easy to see that the derivative $\displaystyle \frac {dx_i} {dt}$ is a rational function of $x_i$, $y_i$ and $t$.

So if $R(x,y)$ is any rational function of $x$ and $y$, then the sum

\begin{equation*} R(x_1,y_1) \frac {dx_1} {dt} \space + \space \ldots \space + \space R(x_n,y_n) \frac {dx_n} {dt} \space = \space S(t) \end{equation*}

is a rational symmetric function of $x_i$ and $y_i$ and is therefore, not just an algebraic function of $t$, but a rational function $S(t)$.

Integrating equation sum then shows the sum of indefinite algebraic integrals (anti-derivatives)

\begin{equation*} \int R(x_1,y_1) dx_1 \space + \space \ldots \space + \space \int R(x_n,y_n) dx_n \space = \space \int S(t) dt \end{equation*}

is simply the integral (anti-derivative) of a rational function $S(t)$.

Equation sum can also be written as a sum of definite integrals. Let $\gamma_i$ be the path described by $(x_i,y_i)$ on the Riemann surface $F(x,y)=0$ when $t$ runs from $t_0$ to $t$ along a path $\tau$ on the punctured Riemann sphere, then

\begin{equation*} \int_{\gamma_1} R(x,y) dx \space + \space \ldots \space + \space \int_{\gamma_n} R(x,y) dx \space = \space \int_{\tau} S(t) dt \end{equation*}

Writing definite less precisely, by giving the bounds of integration rather than the paths, gives

\begin{equation*} \int_{x_1(t_0)}^{x_1(t)} R(x,y) dx \space + \space \ldots \space + \space \int_{x_n(t_0)}^{x_n(t)} R(x,y) dx \space = \space \int_{t_0}^{t} S(t) dt \end{equation*}

Equation definite2 is usually written with a fixed lower bound $x_0$ by adding a constant $C$ independent of $t$ to the RHS, giving

\begin{equation*} \int_{x_0}^{x_1(t)} R(x,y) dx \space + \space \ldots \space + \space \int_{x_0}^{x_n(t)} R(x,y) dx \space = \space C \space + \space \int_{t_0}^t S(t) dt \qquad \text{where} \qquad C \space = \space \int_{x_0}^{x_1(t_0)} R(x,y) dx \space + \space \ldots \space + \space \int_{x_0}^{x_n(t_0)} R(x,y) dx \end{equation*}

Let's look at the quartic curve for example

\begin{equation*} F(x,y) = y^2 - R(x) \qquad\qquad G(x,y,t) = y - t \end{equation*}

where

\begin{equation*} R(x) \space = \space ax^4 + bx^3 + cx^2 + dx + e \space = \space a(x-e_1)(x-e_2)(x-e_3)(x-e_4) \end{equation*}

For definiteness suppose that the $e_i$ are real with $e_1 \lt e_2 \lt e_3 \lt e_4$ and $a$ is real with $a \lt 0$. Then the graph of $F$ is two ovals intersecting the $x$-axis at $e_1, e_2$ and $e_3, e_4$. The curve $G$ is a horizontal line intersecting $F$ at four points, for $|t|$ sufficiently small. We have $x_i(t)$ defined implicitly as the roots of $t^2 - R(x) = 0$ and $y_i = t$. Therefore

\begin{equation*} \frac {dx_i} {dt} \space = \space \frac {2t} {R'(x_i)} \end{equation*}

and (using CAS for the final step)

\begin{equation*} S(t) \space = \space \sum_i T(x_i,y_i) \frac {dx_i} {dt} \space = \space \sum_i \frac {2t T(x_i,t)} {x_i R'(x_i)} \end{equation*}

Integrating from $0$ to $t$, and taking $x_i(0) = e_i$ gives

\begin{equation*} \bigint_{e_1}^{x_1(t)} T(x,y)dx \enspace + \enspace \bigint_{e_2}^{x_2(t)} T(x,y)dx \enspace + \enspace \bigint_{e_3}^{x_3(t)} T(x,y)dx \enspace + \enspace \bigint_{e_4}^{x_4(t)} T(x,y)dx \enspace = \enspace \bigint_0^t S(t) dt \end{equation*}

Specific examples of $T$ are

Table 1
$T(x,y)$ $S(t)$ $\displaystyle \int_0^t S(t) dt$
$\displaystyle \frac 1 y$ $0$ $0$
$\displaystyle y$ $0$ $0$
$\displaystyle \frac y x$ $\displaystyle \frac {2t^2} {t^2-e}$ $\displaystyle 2t + \sqrt{e}\log\left(\frac {\sqrt{e} - t} {\sqrt{e} + t} \right)$

References

[1] Christian Skau Abel and abelian integrals Forum for matematiske perler, Foredrag 2016–2017

[2] Christian Houzel The Work of Neils Henrik Abel Conference: The Abel bicentennial

[3] Phillip Griffiths The Legacy of Abel in Algebraic Geometry 10.1007/978-3-642-18908-1_5.

[4] Michael Roberts A Tract on the Addition of Elliptic and Hyper-elliptic Integrals Google Books