in the theory of value cuid prices. 59 



dA^,: dEj ' ' * dM^^, • dA^,, 'dB^,,'"' V7I4, 

 j?U^ ^U dlJ dl^ dJJ_ dJj 



JaT,' dB7'' ' ' ' dWr, ' TuT, ' dB7, ' ' ■ 'dW. 



(2 7n — 1) n 

 indepen- 



,2 ao^^, am^,, ai^^^o ud^^, am^, 2 1 ^^^^ 



dJJ dJJ dJJ d\J dU dV 



equations. 

 no new 



dA^^,,' c?B^,„' ' ' ' dM^„ ' dA^^„ ' dB^^„' ' ' ' dM^^~ \ unknowns. 



— Pa ' —Pb '• • '' —P,n ■ +Pa ' ^ Pb ' - ■ ^ P ,a J 



l!^o. equations: r}%-\-{7i—\)-\-1inn^{2m—\) n-=:\inn-^m — \ 

 No. unknowns: 2r/zri + ^/i + 2m?i + ■=.^ynn-\-r)\. 



There are just one too few equations. It may not be evident at 

 first why the second set does not contain n independent equations 

 instead of (n— 1). The point is that any one of these equations can 

 be derived from the others together with the equations of the first 

 set. Thus multiply the equations of the first set by jt>,^, ^9j, - - - Pm. 

 respectively and add the resulting equations arranging as follows : 



H- I 



■\- K.r.' Pa-\- B,,„ . Pi + • • • + M^_, . p^ J 



r A^,, . p^, + B^,, .;>, + ... + M,,, . ^„, + 

 I +A^,, . p,, + B^,, . ^9, + . . . + M^ , . p„, + 



L ^-K,n . Pa + B^,, . Pb-V ' ' . 4- M^,„ . p,^ . 



Subtracting from this equation the sum of all but the first (saj'') 

 of the second set, our result is : 



A^, 1 . i^ + B^, , . ^j + . . . + M^,i ^ Pr. ^ 



A,,, . P. + B^,, . ^9^ -I- ... -I- M^,i . /)„ 



which is the first equation of the second set. This equation is there- 

 fore dependent on the others, or there is one less independent equa- 

 tion than appears at first glance. Hence we need one more equa- 

 tion. We may let: 



Pa =1. 



This makes A the standard of value (cf. vi^ 5). 



No such limitation applies to the equations in Chapter IV, 



