( 145 ) 



By calculation and arrangement of the terms of the minimum 

 condition we arrive at 



C\ — 2 [g (aa^ + 6ft, + cy.«)(a«v + 6ft -f- cy v )] — 



— 2 gp. (a^« v -f b,, ft + c, x y v ) — 2 gr v (a* «/« + ^ ft* -f c v 7//) 



The expression |> (a «„. -f 6 ft, + c y„) (a a« + & ft + c 7*)] being put 



equal to F, the conditions for the minimum furnish the following 



equations : 



dF dF dF 



— = g v a v — = g v b v ^— = ^v c», 



oa^ Oft, oy /x 



dF dF dF 



Such an expression F which, as far as the coefficients a, ft y 

 appear in it, contains only products of one of the a /JL ft, y„. with one 

 of the «v ft y v can be written as linear expression in each of those 

 sets of 3 coefficients in the following way : 



OF dF dF dF dF , dF 



d^ dft, d//* d« v Oft oy v 



So that by substituting the equations resulting from the minimum 

 we arrive at the following relations : 



F = g v (av €tp + 6 V ft, -f- cv y^.) = ^ (a^ a v -f 6^ ft + fy yv) • 

 In words this relation runs: with equal weights an error in ^ 

 has equal influence on the apparent error ƒ, as an equally large 

 error in I has on the apparent error ƒ„. If the weights of the two 

 results of measuring are unequal, errors in these which are in inverse 

 ratio with their weights will cause each other's apparent errors to 

 deviate to the same amount from the true ones. 

 Let us put in the condition 



[g {I — a [al] — b [#] — c [yf] } a ] = minimum 

 li = 1 and all other quantities / = , then from this arises 



gi (1 — 2a,«,- — 26,-ft — 2ciy,) + [g (acti + 6ft -f cy,) 1 ] = min. 

 from which we deduce putting [g {aai f 6ft + cy«) 5 ] = G : 



°G n dG n z dG o 

 — - = 2g iai — = 2^rï6« — = Igxd 

 dcti oft oy,- 



and from this ensues again : 



1 dG 1 dö 1 dG , , . _ 



Ö = «,- + _ ft— + -y,— = gi {ohh -f 6,-ft + W* 



2 o«i 2 o/?,- 2 oy,- 



With the aid of the above deduced theorem each term of the 

 summation in the expression [#(««,- + 6 ft -f c y,- )'] can be replaced 



10 

 Proceedings Royal Acad. Amsterdam. Vol. X. 



