62 



Irving Fisher — Mathematical investigations 



_^U_ _^riJ_ ._^^.^_. dJJ _^ n{?,m-l) 



d\J_ 



d\j_ ,J^ d\J_ ^dV_ jdV dJJ 



dA. 



independ- 

 ent equa- 

 tions, 



no new 



un- 

 knowns. 



Pa, IT • ' ' 'Pm, TT ' Pa, e • • • • Pm, g • /\, k * ' ' ' Pm, k 



Pa,^ + Pa,, —Pa.^ ) '^^ equations. 



,, _[_ ^0 ^0 ) no new unknowns. 



No. equations: 2ni •^(n — \)-\-^m7i'{-7i[^')n — l)-\-inz=:Qm7i-\-^tn — \. 

 No. unknowns: 3m?^-|-3m +:^;?^?^-f-0 +0 z^Qmn-\-^7n. 



The second set apparently contains oi equations instead of n—\ 

 as above recorded. But, by multiplication of the first line of the 

 first set, we have : 



(A^,,+ . . . H-A^.„) 2\,n = (A„i+ . . . +K,n) Pa,^ 

 (A^,i-f . . . +A^,„) 2^^,, = iA„,+ . . . +A^,^)p,^^ 



adding and remembering that ^^a,^. ^^^ Pa,Tr~^l'^a,t ^^^ S®^ • 



^., 1 ■ Pa,n + • • • -\- K,n-Pa,^ •\- K ^ ' Pa,e + • • • + ^.,n'Pa,,=^ 



Writing the similar equations from the second, third, etc. lines of the 

 first set and adding we get (rearranging terms) : 





!^ = 



I +A,,o. 2^.+ - - 



^ + A^ „ . 2\.+ . . . +M^,„: ^v. 



If from this equation the sum of all but one of the second set be 

 subtracted the result will evidently be the remaining one. 

 We are therefore at liberty to write 



Pa„ = 1 



to determine a standard of value. 



