= Solution
Let $e$ be the vector of ones. For a general nonautonomous <Runge-Kutta method> the stages satisfy $k_i=f(t+c_ih,y+h\sum_ja_{ij}k_j)$ and the update is $Y=y+h\sum_ib_ik_i$. Expansion at $(t,y)$ gives
$$
k_i=f+h\{c_if_t+(Ae)_if_yf\}+O(h^2),
$$
and therefore
$$
Y=y+h(b^Te)f+h^2\{(b^Tc)f_t+(b^TAe)f_yf\}+O(h^3).
$$
The exact solution is $y(t+h)=y+hf+h^2(f_t+f_yf)/2+O(h^3)$. Matching the independent terms gives the necessary and sufficient <second-order conditions with independent Runge-Kutta abscissae>
$$
\boxed{b^Te=1,\qquad b^Tc=\tfrac12,\qquad b^TAe=\tfrac12.}
$$
Under the usual internally consistent <Butcher tableau> convention $c=Ae$, this reduces to $c=Ae$, $b^Te=1$, $b^Tc=1/2$. If the abscissae are allowed to be independent of the row sums, $c=Ae$ is a sufficient simplifying convention rather than a logically necessary order-two condition.
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