220 Professor Whewell on the Mathematical Exposition 



c„-i c„_, c„ ' p r„c„_i c„^ipr„ 



Let as before 



Tn^ilSih = 1 +p^ -^ =1, r = mr„ ; and we have 



(1 +«))(! +/>) = !+—. 



On the same suppositions as those ah-eady made in Art. 22, 

 we shall have, as there, 



p = m {fi — 1) u, w = eu; whence 



7 fit 



(1 +eu) [l+m(^-l)«} - 1 + — ; 



and hence, as before, omitting m", &c. 



7nt .-, t , 



pr\e + m{ix—\)\ pr 



ink 

 u = 



e + »t(,u - 1) ' 

 k is the fraction when the tax t is of the average produce. 



Also as above, cqv = rpu, a„ = mua. 



Substituting these values in the expression above given, and 

 also w for eu, and kipr for I, we have 



_ tR= —arp {{e — l)u — eur\ + arpk{\ - mu) — arpu 

 = arpk (1 —mu) - arpeu (1 — u) 



= arp \k - eu— {ink — eu) u]. 



And taking the value of u above found, this gives 



_^R^ arpk { ^K^^-l)-e(>«-l) _ , "I'^^-'^'J . 

 ' I e + m(/i— 1) {e + m(M — 1)}J 



If, as before, we suppose e = 3, m = 4, n = 7> the two terms 



within the brackets become - and - -■ If A- be small, as— > 



y 7-'" ^^ 



the latter term is small in comparison of the first. 



