210 Professor Whewell on the Mathematical Exposition 



quality. Also let r„_, be the rate of produce of the next quality, 

 and a„_i its quantity. When a„ is thrown out of cultivation, the 

 produce diminishes from ar to ar - a„r„ or ar{l-u): hence the 

 price increases from p to //. Suppose />'=(! +tv)p, then since 

 the increase of price depends on the diminution of supply, to is 

 a function of u as has already been explained. 



In the case in which limiting soils determine tlie price, since 

 p is the limiting price when the lowest rate of produce is r„, and 

 ;/ when it is r„-i, with a tax <„_i: we shall have by Axioms 5 

 and 6, jnitting t„_, = k„_, p'r„_^, 



P'r„-i{i - A„-i) = ?c„_„ 

 PK = qc„, 



P' 

 and dividing, observing that — - \ + w. 



>■„_, c„ 



(1 +«;)(! -/,„_.) -p=^ (a). 



the quantities -"— and ^^-^ depend upon u, the diminution of pro- 

 duction, if Ave suppose the scale of soils, that is, the fertility and 

 (juantity of each quality, and the capital requisite for its culti- 

 vation, to be known. On this supposition, all the quantities in 

 the preceding equation (a) would be functions of u, and hence 

 the equation would serve to determine u, and we should rind 

 the quantity by which the produce was diminished. 



But without attempting this exact solution of the problem, 

 we shall be able to obtain approximations sufficiently general 

 and accurate. 



If we suppose the capital employed on an acre each of the 

 last two qualities to be the same, and the produce only to be 

 difterent ia the ratio i + /o : l , we have 



(1 ■\-w){\ -fr„_,)(l +^) = 1. 



