= Solution
For <PP chain> burning and the <Kramers' opacity law>, insert $\eta=4$, $\lambda=1$, and $\nu=-7/2$ into the <homology relations for radiative stars>:
$$
X=\frac{1/2}{13/2}=\frac1{13},\qquad
Y=6-7X=\frac{71}{13}.
$$
Thus the <radiative homology with proton-proton burning and Kramers opacity> is
$$
\boxed{R\propto M^{1/13},\qquad L\propto M^{71/13}.}
$$
To obtain the <Hertzsprung-Russell diagram> slope, apply the <Stefan–Boltzmann law> $L=4\pi R^2\sigma T_{\rm eff}^4$:
$$
T_{\rm eff}\propto M^{(Y-2X)/4}=M^{69/52},\qquad
\boxed{\frac{d\log L}{d\log T_{\rm eff}}=\frac{4Y}{Y-2X}
=\frac{284}{69}\simeq4.116.}
$$
Equivalently $T_{\rm eff}\propto L^{69/284}$. The <Hertzsprung-Russell slope of a stellar homology sequence> refers here to logarithmic <luminosity> versus logarithmic effective <temperature>; a conventional diagram reverses its horizontal axis, so the line descends toward the right.
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