Suppose a complex function $f(z)$ has a complex derivative at $z_0$. If we write the real and imaginary parts of $f$ as $u+iv$ and those of $z$ as $x+iy$ (so $u$ and $v$ are functions of $x$ and $y$) then $$\begin{array}{c} {\displaystyle \frac{\partial u}{\partial x} = \frac{\partial v}{\partial y} }\\ ~\\ {\displaystyle \frac{\partial u}{\partial y} = -\frac{\partial v}{\partial x}}. \end{array}$$ These are now called the Cauchy-Riemann equations. They appear in a slightly different form in Cauchy's 1814 work Mémoire sur les intégrales définies, reprinted in his Oeuvres complètes Ser. 1, 1, Paris (1882), on page 338.
Cauchy starts by defining an indefinite integral $F(y)$ with $F'(y) = f(y)$, where $y$ is a function of $x$ and $z$. Then $$\frac{\partial F}{\partial x} = f(y)\frac{\partial y}{\partial x}$$ $$\frac{\partial F}{\partial z} = f(y)\frac{\partial y}{\partial z}. $$ Taking mixed second partials we have $$\frac{\partial }{\partial z} [f(y)\frac{\partial y}{\partial x}] = \frac{\partial }{\partial x} [f(y)\frac{\partial y}{\partial z}]. ~~~(1)$$
Cauchy now supposes $M$ and $N$ are arbitrary functions of $x$ and $z$, with $y=M+iN$ (Cauchy writes $\sqrt{-1}$) and that $f(M+iN) = P' +iP''$. Then equation (1) becomes $$\frac{\partial }{\partial z}[( P' +iP'')(\frac{\partial M }{\partial x} + i \frac{\partial N}{\partial x})] = \frac{\partial }{\partial x}[( P' +iP'')(\frac{\partial M }{\partial z} + i \frac{\partial N}{\partial z})] ~~(2),$$ or $$\frac{\partial }{\partial z}[( P' \frac{\partial M }{\partial x}-P''\frac{\partial N}{\partial x}) +i(P''\frac{\partial M }{\partial x} + P'\frac{\partial N}{\partial x})] = \frac{\partial }{\partial x}[( P' \frac{\partial M }{\partial z}-P''\frac{\partial N}{\partial z}) +i(P''\frac{\partial M }{\partial z} + P'\frac{\partial N}{\partial z})].~~~(3)$$ Cauchy writes: $$\begin{array}{cc} {\displaystyle P' \frac{\partial M }{\partial x}-P''\frac{\partial N}{\partial x} = S} & {\displaystyle P' \frac{\partial M }{\partial z}-P''\frac{\partial N}{\partial z}=U}\\ ~\\ {\displaystyle P'\frac{\partial N}{\partial x}+ P''\frac{\partial M }{\partial x} =T} & {\displaystyle P'\frac{\partial N}{\partial z} +P''\frac{\partial M }{\partial z} = V}. \end{array}$$ Then equation (3) becomes $$\frac {\partial S}{\partial z} + i\frac {\partial T}{\partial z} = \frac {\partial U}{\partial x} + i\frac {\partial V}{\partial x}.$$ [He notes that if instead of $y=M+iN$ we had taken $y=M-iN$, we would have found $$\frac {\partial S}{\partial z} - i\frac {\partial T}{\partial z} = \frac {\partial U}{\partial x} - i\frac {\partial V}{\partial x}.]$$ Separating real and imaginary parts, this gives $$\begin{array}{c} {\displaystyle \frac{\partial S}{\partial z} = \frac{\partial U}{\partial x} }\\ ~\\ {\displaystyle \frac{\partial T}{\partial z} = \frac{\partial V}{\partial x}}. \end{array}$$ Cauchy remarks: "These two equations completely contain the theory of the transition from real to imaginary."
To retrieve today's Cauchy-Riemann equations one takes $M=x,~N=z$. Then equation $(2)$ becomes $$\frac{\partial }{\partial z}[( P' +iP'') (1+0)] = \frac{\partial }{\partial x}[( P' +iP'')(0+i)],$$ or $$ \frac{\partial P' }{\partial z} + i \frac{\partial P''}{\partial z} = i\frac{\partial P'}{\partial x} - \frac{\partial P''}{\partial x}.$$ Hence $$\begin{array}{c} {\displaystyle \frac{\partial P'}{\partial x} = \frac{\partial P''}{\partial z} }\\ ~\\ {\displaystyle \frac{\partial P'}{\partial z} = -\frac{\partial P''}{\partial x}}. \end{array}$$ Writing $P'=u, ~P'' = v$ and $z=y$ puts them in their familiar form.