*DD*' being the demand curve, and

*SS*' the curve corresponding to the supply schedule described in the text, let

*MP*

_{2}

*P*

_{1}be drawn vertically from any point

*M*in

*Ox,*cutting

*SS*' in

*P*

_{2}and

*DD*' in

*P*

_{1}; and from it cut off

*MP*

_{3}=

*P*

_{2}

*P*

_{1}, then the locus of

*P*

_{3}will be our third curve,

*monopoly revenue curve.*The supply price for a small quantity of gas will of course be very high; and in the neighbourhood of

*Oy*the supply curve will be above the demand curve, and therefore the net revenue curve will be below

*Ox.*It will cut

*Ox*in

*K*and again in

*H,*points which are vertically under

*B*and

*A,*the two points of intersection of the demand and supply curves. The maximum monopoly revenue will then be obtained by finding a point

*q*

_{2}on

*Lq*

_{3}being drawn perpendicular to

*Ox, OL*×

*Lq*

_{3}is a maximum.

*Lq*

_{3}being produced to cut

*SS*' in

*q*

_{2}and

*DD*' in

*q*

_{1}, the company, if desiring to obtain the greatest immediate monopoly revenue, will fix the price per thousand feet at

*Lq*

_{1}, and consequently will sell

*OL*thousand feet; the expenses of production will be

*Lq*

_{2}per thousand feet, and the aggregate net revenue will be

*OL*×

*q*

_{2}

*q*

_{1}, or which is the same thing

*OL*×

*Lq*

_{3}.

The dotted lines in the diagram are known to mathematicians as rectangular hyperbolas; but we may call them *constant revenue curves:* for they are such that if from a point on any one of them lines be drawn perpendicular to *Ox* and *Oy* respectively (the one representing revenue per thousand feet and the other representing the number of thousand feet sold), then the product of these will be a constant quantity for every point on one and the same curve. This product is of course a smaller quantity for the inner curves, those nearer *Ox* and *Oy,* than it is for the outer curves. And consequently since *P*_{3} is on a smaller constant revenue curve than *q*_{3} is, *OM* × *MP*_{3} is less than *OL* × *Lq*_{3}. It will be noticed that *q*_{3} is the point in which *QQ*' touches one of these curves. That is, *q*_{3} is on a larger constant revenue curve than is any other point on *QQ*'; and therefore *OL* × *Lq*_{3} is greater than *OM* × *MP*_{3}, not only in the position given to *M* in the figure, but also in any position that *M* can take along *Ox.* That is to say, *q*_{3} has been correctly determined as the point on *QQ*' corresponding to the maximum total monopoly revenue. And thus we get the rule:—If through that point in which *QQ*' touches one of a series of constant revenue curves, a line be drawn vertically to cut the demand curve, then the distance of that point of intersection from *Ox* will be the price at which the commodity should be offered for sale in order that it may afford the maximum monopoly revenue. See Note XXII. in the Mathematical Appendix.