226 Mr. R. Meldrum Stewart on the 



We now proceed to the solution of the problem of de- 

 termination of density from measurements of a mass and 

 a volume, in which case Dr. Campbell states * that the 

 method of least squares is clearly inapplicable. The mass (jc) 

 and the volume (y) of a number of different samples of 

 a substance are supposed to have been measured. Then 

 if b be the density, the general condition equation con- 

 necting the measured quantities is x = by. It is required 

 to find the most probable value of b. 



Eliminating a? ls # 2 , x z ... by substituting their values 

 from the condition equations, and putting 



b = B + A6, 



Vi = Yj + At/!, 

 i/ 2 = Y 2 + A?/ 2 , 

 etc., 



the observation equations become 



yi = yi B 2/l +Y 1 A6 = t T 1 ', 



y 2 = ?// B?/ 2 + Y 2 A6 = a? s ', 



Assume that x-l, x 2 . . . are measured with weight unity, 

 and yi, y 2 • • • with weight p ; then the normal equations 

 are 



(B' + pfa* BY X M =Bx l '+py 1 ' 

 (B 2 +p)y 2 + B Y 2 A6 = Bx 2 ' +^ 8 ' 



B^Yz/ + 2Y 2 .A£ = 2^'Y. 



We may now make use of a simple device to obviate the 

 necessity of successive approximations. Assume that the 

 approximate values B, Y 1? Y 2 . . . have been chosen exactly 

 correctly : then necessarily A6 = 0, B = 6, Y 1 = y 1 , Y 2 =y 2 , 

 etc., and the normal equations become 



(b 2 +p)y l = bx{+py^ 



(b 2 +p)y 2 = bx 2 ' + V y 2 r 



b.2y 2 = %x'y. 

 * Loc. cit. p. 192. 



