Pagina:Scientia - Vol. VII.djvu/106

(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, ${\displaystyle x={\frac {1}{\sqrt {2}}}\mathrm {R} }$, ${\displaystyle y={\frac {1}{\sqrt {2}}}\mathrm {R} }$. 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, ${\displaystyle \scriptstyle {\sqrt {2}}}$ 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 ${\displaystyle {\frac {1}{\sqrt {2}}}\mathrm {R} }$ to ${\displaystyle {\frac {1}{\sqrt {2}}}\mathrm {R} -0.01\mathrm {R} }$ the ordinate remaining unchanged is found to be
${\displaystyle \scriptstyle \mathrm {R} -{\sqrt {\mathrm {R} ^{2}-0.0001\mathrm {R} ^{2}}}=\mathrm {R} \left(1-{\sqrt {0.9999}}\right)}$.
Likewise the diminution of the ordinate by ${\displaystyle \scriptstyle {\sqrt {2}}}$ per cent thereof is ${\displaystyle \scriptstyle \mathrm {R} \left(1-{\sqrt {0.9999}}\right)}$. The sum of these two effects, viz. ${\displaystyle \scriptstyle 2\mathrm {R} \left(1-{\sqrt {0.9999}}\right)}$ is to be compared with the effect of taking the two doses together, viz. ${\displaystyle \scriptstyle \mathrm {R} \left(1-{\sqrt {0.9998}}\right)}$. 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.