280 



DR JAS. BURGESS ON 



22. In the theory of Errors of Observations, we may state the proportions of 

 the different constants for 

 1 modulus,' ' mean error,' 

 * error of mean square,' 

 and ' probable error,' as 

 in the adjoining table. # 

 And the ordinary rela- 

 tions of ' mean' or average 

 error, A (double the mean 

 risk) ; weight of an obser- 

 vation, or square of the 

 number of observations 

 divided by twice the sum 

 of the squares of the errors, 

 W ; modulus, M ; the error 

 of mean square, S ; and probable error, E, — are expressed by the equations, — 





Modu- 

 lus. 



Mean 

 Error. 



Error 

 of Mean 

 Square. 



Prob- 

 able 

 Error. 



In terms of modulus 



1 



1 



1 



x/2 



P 



pjir 



In terms of mean error 



J* 



1 



/- 

 V 2 



In terms of error of mean square 



/2 



1 



ptf 



In terms of probable error 



i _L 

 P pJt 



1 

 P J2 



1 



M = Aj7T = SJ2: 



E 



JW 



E = Mp = A P Jw = S P J2 = £ 



W 



M 2_ 7rA 2 2S 2 E2 



A * 2 E 1 



~ J2~ A ^ 2-^2- V2W 

 _ 1 _P 2 



,(39) 



The values of the constants are found in the table above ; but for approximations 

 that are occasionally useful, the following may be given : — 



296 , -^ (ce ,, 709 



For Jtt we may use ^= = 1-772 455, or roughly ^; 



for 



4 ■ ■ 



7T 



for 



J2 „ „ 



for 



p 



for 



pJ2 „ „ 



for 



P Jtt » » 



and for p 2 „ „ 



500' 



^ = 0-797 8848, or g| = 797 895, or g= 797 87, or 3 ' 

 g| = 1-414 2156, or g|= 1414 201, or ?? = 1-414 286 ; 

 g-|| = 0476 939, or|^= -476 9475, or f -476 923; 



|JU = 0-6744898, or |^= -674497, org '67445 



65 

 29 



43 : 



^| = 0-845 3472, or g = -845 36, or g= "845 24; 

 ^ = 0-227 468, or^-= -227 488, or^= -227 27; 



„ 296. 239 „ 99 a 629 _ 65 1 



Whence,- M = ^A = m S,or^S= ^E, ov^E--^ . 



* Conf. Airy's Theory of Errors, p. 24 ; Galloway's Treat, on Probability, §§ 145-148, pp. 194-197 ; De Morgan 

 Essay, p. 139. 



