Pagina:Scientia - Vol. VII.djvu/106

98

“scientia„

(concave) surface of a spherical — or rather hemispherical — bowl underneath, the rim of which is the circle shown in the figure (being a «great circle» of the sphere of which the bowl forms a hemisphere). Thus initially when x and y are each zero, z is zero. As x and y are increased z will increase up to the maximum which occurs at the point P; for which if R is the radius of the sphere and of the (great) circle shown in the figure, $x={\frac {1}{\sqrt {2}}}\mathrm {R}$, $y={\frac {1}{\sqrt {2}}}\mathrm {R}$. Now let us consider the effect of removing — first successively, then simultaneously — small doses of each factor, say one per cent of each, or as more convenient to calculate, while less favourable to our thesis, $\scriptstyle {\sqrt {2}}$ per cent of each that is 0.01R. Whereas R measured the maximum net profit, the diminution of profit caused by reducing the abscissa from ${\frac {1}{\sqrt {2}}}\mathrm {R}$ to ${\frac {1}{\sqrt {2}}}\mathrm {R} -0.01\mathrm {R}$ the ordinate remaining unchanged is found to be

$\scriptstyle \mathrm {R} -{\sqrt {\mathrm {R} ^{2}-0.0001\mathrm {R} ^{2}}}=\mathrm {R} \left(1-{\sqrt {0.9999}}\right)$.

Likewise the diminution of the ordinate by $\scriptstyle {\sqrt {2}}$ per cent thereof is $\scriptstyle \mathrm {R} \left(1-{\sqrt {0.9999}}\right)$. The sum of these two effects, viz. $\scriptstyle 2\mathrm {R} \left(1-{\sqrt {0.9999}}\right)$ is to be compared with the effect of taking the two doses together, viz. $\scriptstyle \mathrm {R} \left(1-{\sqrt {0.9998}}\right)$. It appears that while the former subtrahend is 0.0001000025..R, the latter is 0.000100005..R — not a very important difference, per cent of net profits (on gross receipts). The difference will still be insignificant even when we take away doses so large as 0.1R, that is above fourteen per cent of each factor. The sum of the effects of (removing) the two doses separately is now 0.010025..R; the effect of the two doses together is 0.01005..R.

Of course if we go on increasing the size of the doses, we shall reach a case in which the difference under consideration is significant. Thus if all industry were organized on a scale in which no entrepreneur could take on more than one or two employées, the remuneration of the last employée taken on would very imperfectly measure — might afford only, a very superior limit to — the remuneration of an employée. In fact that remuneration would be indeter-