100 



Another division of commodities is, according to the cost of pro- 

 duction. These are (a) commodities of fixed and limited supply ; 

 (/3) commodities of fixed cost ; (y) commodities of increasing cost 

 for increasing supply, as for instance corn in a given limited district. 

 The equation of price for the last case was given. 



The like methods were applied to solve certain problems concern- 

 ing international trade, treated by Mr. Mill. If the relative value 

 of two commodities, C and D, in England and Germany be different, 

 there will be a saving in exporting each from where it is cheaper to 

 where it is dearer ; and the question is, at what point prices will 

 settle. We must introduce here the principle of the uniformity of 

 international prices ; namely, that when the trade is established, the 

 relative prices of C and D will be the same in the two countries : 

 the principle of the equality of imports and exports in each country ; 

 and the equation of demand and supply already stated. By combining 

 these principles, the problem of the resulting price is solved. But 

 it is found that there is no solution possible (that is, no solution in 

 which both countries gain by the trade), except the mutual demand 

 for the interchange of commodities be nearly equal. This limitation 

 of the solution is given by the algebraical method, and seems to have 

 been overlooked by previous writers. 



The same methods were extended to a greater number of ex- 

 ported and imported commodities ; and finally, it was remarked that 

 these calculations are all founded on principles of equilibrium, 

 whereas a state of equilibrium is never attained ; and thus the theory 

 may be very imperfectly applicable, like the equilibrium theory of 

 the tides. 



Second Memoir on the Intrinsic Equation of Curves. By the 

 Master of Trinity. 



The intrinsic equation of curves, according to which any curve is 

 expressed by means of an equation between its length (s) and its 

 angle of deflection {<p), may be conveniently used for many purposes. 

 When a curve is so represented, the portion of the length which 

 comes after a cusp must necessarily be taken as negative. This had 

 appeared anomalous to some mathematicians, on the ground that a 

 cusp is in all cases the limit of a loop. To clear up this point, the 

 author adduces two cases. (1.) The curve of which the equation is 

 s=a<p-\-b sin <p, which is a looped curve when b is less than a, and 

 a cusped curve otherwise. But in this curve it appears that a loop 

 arises from the vanishing of two cusps, and of the intervening nega- 

 tive portion of the arc. (2.) The case of the ordinary trochoid, which 

 is a looped curve when the describing point is exterior to the rolling 

 circle, and becomes a cusped curve (a cycloid) when the point is in 

 the circle. But in this case the length of the trochoid is equal to 

 the length of an elliptical arc, which, in the case of the cycloid, 

 coincides with the major axis, and becomes negative beyond the 

 vertex of the ellipse. Other equations were examined, which give 

 running pattern curves with cusps, cusped curves with infinite diver- 

 ging spirals at the extremities, and sinuous curves with infinite con- 

 verging spirals at the extremities ; and certain integrals which oc- 

 curred in the former memoir on this subject were discussed. 



