144 Dr. WHEWELL, ON THE MATHEMATICAL EXPOSITION OF 



52. But we must introduce an additional consideration. The relative price of (C) and 

 (E), both of which are produced in England, will depend on their relative cost in England, and 

 will not be altered by the trade. And, in like manner, the relative price of (D) and (F), which 

 are produced in Germany, will depend upon their relative cost in Germany, and will not be 

 altered by the trade. This we may term the principle of the relation of cost in the pro- 

 ducing country. 



Hence the ratio of r to p' is the same as the ratio of r to p. Let r = yp, then r' = yp'. 

 In the same way if R = TP, R 1 = YP". And hence j? is the same for p' and r' ; X, for 

 P' and R'. 



53. The quantity s of (F) is equivalent, in Germany, to IV of (D) : hence the whole 

 import of England is equivalent to q + TV of (D), which in terms of (C) is p'q' + Tp's'. The 

 export of England is Q' of (C) and & of (F), equivalent to Q' + yS' of (C). Hence the equa- 

 tion of import and export, (37), becomes 



p'q' + Fp's' = Q' + 7-S^. 

 In like manner the equation of import and export for Germany is 



q + Ts' = P'Q' + yP'S'. 

 Hence, if the equation of supply and demand for (E) and (F) (22) be respectively, 



r's' = rs (1 - nx), R'S' = RS (X - NX), 

 we shall have 



q 1 ~ mx + rs 1 — n - = PQ(l - MX) + yPS(\ - NX). 



But we have, as before, under the trade, by the uniformity of international prices, (36) 



P'p' = 1. And if. as before, — — = 1 - k, 



Pp 



(l - X) (1 - w) m l - *, whence X = . 



1 — 00 



Hence 



q(\-mx) + T*(l -m)= PQ{l - Mk-(\ - M)x\ + y PS\l - NK - (l -N)»\, 



W hence . = j_ + F« - «l - JfAQ PQ + (1 - Mfry iff 

 mq + nrs - { (l - M)PQ + 7 (l - N)PS\ 



If we suppose that n = m and N = M, this becomes 



q + Ts - (l - Mk)P(Q + ys) 

 m(q+ Vs) - (1 - M) P(Q + ys) ' 



And if P(Q + ys) = n'(q+ Ts), 



1 -» w'(l - il/A) 

 W = m-»'(l - it^) ' 



as before (39). And the applicable values of oo will occur only for values of n' between certain 

 limits, as before. That is, the trade will be profitable to both countries only when the 

 inequality of mutual demand is within certain moderate limits. 



