62

Irving Fisher ^-Mathematical investigations

d UdA^ :c?U

dA w.* ‘

■ ‘ dM ni , ' dA ti ,dTJ dTJdA t $

dXJ

dTJ

■ ■■

dTJdA K , s

d\J

■ 1 dM K , \ -

_

n(Zm —1)independ-ent equa-tions.

r

«!U

dTJ -tfU

dTJ

dTJ

dTJ ~

no new

un-

dK,a ’

"dM ni „' dA"„

' dA *,«"

"rfM.,.-

knowns.

Pa,„ ■

■ -Pm,„ ■ Pa„

* Pal, t

Pa, k' '

: - Pm, K

Pa, „ + Pa

P«,*+ Pm,

[

e = Pm, a

m equations.

no new unknowns.

No. equations : 2w* -\-(n— l)-\-Zmn+n(Zm— \)-\-m=.Q>mn-\-Zm — 1.No. unknowns: Zmn-\-Zm -\-Zmn-\-0 -)-() =6mn-j-3m.

The second set apparently contains n equations instead of n— 1as above recorded. But, by multiplication of the first line of thefirst set, we have :

( A w ,i+ • • • + A TT,n) Pa, n ( A «,> + • • • + A K ,n) Pa.rr

( A «,l + • ' ’ + A e,») Pa,a ~ l A *,l + ' • • +A K ,n)Pa, t

adding and remembering that p„ iK = Pa,„+Pa„ we get :

A „, 1 • Pa, v + • • • ■+■ A tt, n • Pa, „ 4" A e, 1 ■ f’a, e + • • • + A e , n • Pa, e =

A (t. 1 • Pa, *+ • • • + A k,« • Pa, k

Writing the similar equations from the second, third, etc. lines of thefirst set and adding we get (rearranging terms):

A t, 1 • Pa,„ + • • + I . p m ,„ 4- A e ,, . />„, r + . . + , . p m< c -j-

+ A :r, 2- Pa, * + • • + !•/>»,„+ .-.-. _ _

+ A t,« • Pa,„ +..+ ■ Pm, t

A „1 • Pa, K + ■ ■ • + M.,, . p miK ++ A «, » • Pa, K +.

k "4" A «, » ‘ Pa, k “I" • ' • “H n • Pm, K

If from this equation the 'sum of all but one of the second set besubtracted the result will evidently be the remaining one.

We are therefore at liberty to write

Pa, K = 1

to determine a standard of value.