I've been reading the economist Francis Ysidro Edgeworth's 'Old and New Methods of Ethics' (1877) and trying to understand his usage of the 'Calculus of Variations' (COV) for the purposes of articulating mathematically a certain view of utilitarianism that he is discussing. What I'm having trouble understanding is how any of these computations work. I have a bit of knowledge of the COV but some of the notation and steps in reasoning he's making are impossible for me to make sense of. He writes as follows:
"Case I. Where the sensibility and capacity are constant throughout the given tract.
The first term of the complete variation of $\int_{x_0}^{x_1} k\{f(y) - f(\beta)\} - cy dx$ is $\int_{x_0}^{x_1}(kf^1 (y) - c)\delta y dx^1 + dx_1 x\{k|f(y) - f(\beta)| - cy\}_1 - dx_0 x\{k|f(y) - f(\beta)|- cy\}_0;$ which vanishes if (1) $kf^1 (y) - c = 0$, and therefore $y$ is constant; (2) $k\{f(y) - f(\beta)\} - cy = 0$. The second term of the complete variation, when $kf^1(y) - c = 0$, becomes $\dfrac{1}{2} \int_{x_0}^{x_1} k f''(y).\delta y^2 dx;$ which is negative, since by hypothesis $f''(y)$ is negative for all values of $y$ (with which we are concerned). A maximum, therefore, is afforded by the equations (1) and (2). Combining them with the equation $(x_1 - x_0).y = D$, and eliminating $c$ and $y$, we find the $extent$ of the favoured region, $x_1 - x_0$. Its $position$ in the given tract is, as might have been $a \ priori$ expected, indeterminate. Thus the favoured region is $limited$ in extent, indeterminate in position; and the law of distribution is equality."
To clarify, it is possible that some kind of Optical Character Recognition software was used in the rendering of the pdf version of the book I have, which would explain some of the confusing symbolism such as exponents equalling 1 when they might plausibly be interpreted as 1st derivatives, but I highly doubt it seeing as this .pdf appears to be from an original printing (see the photo below if you can). This just adds further to my confusion as you can imagine.
