non-constant variance of $\hat{\epsilon}$ against $\hat{y}$ In Faraways Linear Models (2ed.) page 77, he mention: "when non-constant variance is seen in the plot of $\hat{\epsilon}$ against $\hat{y}$", a transformation of the response $y$ to $h(y)$ where $h()$(a comment on what h() means here is appreciated) can be chosen so that $\operatorname{var}(h(y))$ is constant."
Then he gives an example on how to consider $h$
$$h(y) = h(Ey) + (y-Ey)h'(Ey)+ \cdots \\ \operatorname{var}(h(y)) = 0+h'(Ey)2\operatorname{var}(y)+ \cdots$$
We ignore higher order terms. For $var(h(y))$to be constant we need
$$h'(Ey) \propto (\operatorname{var}(y))^{-\frac{1}{2}}$$
which suggests:
$$h(y) = \int\frac{dy}{\operatorname{var}(y)} == \int\frac{dy}{\text{SD}y}$$
(a comment on what the '$==$' means here is appreciated).
By an example with R, how exactly would this transformation plan out on data with non-constant variance?

The very next sentence in Faraway gives you two examples:

For example if $\operatorname{var} y=\operatorname{var}\varepsilon\propto(Ey)^2,$ then $h(y)=\log y$ is suggested while if $\operatorname{var}\varepsilon\propto(Ey),$ then $h(y)=\sqrt{y}.$

Furthermore, on page 219 you can see an example of the logarithm transformation done in R:
lmodu <- lm(log(longevity) ~ activity, fruitfly)

I imagine the $==$ just means "is equivalent to", similar to $\equiv$ in logic.
