in the Calculation of Errors. 109 



multiplying each of the proportional coefficients above written 

 by A, the determinant of the quantic 



ax 2 + bf + cz* + 2fyz + fyzw + 2%. 



The relative error of the divisor -r has the following elegant 

 expression : — 



2[fe 2 3plus ge % \ plus he^\ . 



The relative error — or briefly r.e. — of any of the prin- 

 cipal coefficients a 



(a 1\ a 1 



— r.e. \ -r-^-r ) = >'. e. -r- minus r.e. -r 



\A A/ A A 



= (since ^ = 1 — /> 2 3 2 ) 2/?i2% mw ne. ^ 



= (by the formula of last paragraph) 



2 U/+ — Bliy 2S pl us g en plus he l2 j . 



If the p's are determined as above on p. 104, the a element 

 of error disappears altogether, and the y element in part. Ac- 



-dingly, putting f-\ — — =k, and expanding, we have 



cor 



r.e. a=2\/ 7 L k s/l—p^ plus g \fl— p n 2 plus h \/l— p 12 2 

 nearly. 



Put as approximate values pi 2 ='8, p 2 3 = '8, p 3l :='9 ; and 

 evaluate the coefficients a, b . . . h. They are approxi- 

 mately a = c = 6, /=/i=l, b — 2>j g = 4=. Substitute these 

 values in the expression for r.e. a ; and we have, when w = 300, 

 modulus of r.e. a= about *5. The probable error for the 

 difference between two determinations = # 5 x *477 x V2 = '33, 

 nearly. 



The relative errors of the other coefficients may be similarly 

 determined, and may be expected to be equally precarious 

 while the number of observations is limited to 300. 



This anticipation is fully borne out by the following ex- 

 perience. From each of the four sets of p's above cited there 

 has been calculated * the exponential quantic of the second 

 degree, or ellipsoid of equal probability. The comparative 

 results are exhibited in the following Table. 



* By Mrs. Bryant. 



