of some Doctrines of Political Economy. 215 



c q 



But pr„ = c„q; .-. »„ = ^; 



.-. /? > p' {l-k), or > /)(1 +«))(!- Ar), 

 and (1 +w) {l-k) < 1. 

 If the tax be not a very large fraction, for instance, if k be 

 ^, or less, we may neglect its powers; 



.•. 1 +w < r J ov < I + k; .-. tc < k, 



1—k 



hence also we may neglect the powers of w, and the products wk, 

 &c. Also by equation {b), we have generally 



{l+w){l-k)=^ 



P 



.•. p = k — w, neglecting powers of k, w ; and therefore p is also small. 



Also we have supposed by Axiom 4, w to be =eu, and hence u 

 also is small. 



26. Expanding equation [d), Art. 23, we have 



1 + {e + m (m-1)m} +e/w(n- 1)m' = JZ^' °'^' P""'"S 7^ =-/' 



e+OT (m- 1) / 



em (fi - 1) em (m- 1) 



And hence 



e + wi( / x-l) ^ ^ e + /»(M-l) f J _^ 4/em(^-l) U 

 2em{^-l) - 2e?n{im — l) I {e+?« (,u- 1)}'J ' 



expanding to three terms of the binomial 



^ ^ e + m (m- 1) ^ f f'emi^-l) 



- 2em{n-l) - e + m(n—l) {e + 7n{fA-l)Y' 



and taking the upper sig-n 



„_ / /'c»i(m-1) 



c + m(m-l) {e + 7M(/ii- I)}*' 



