374 On Discordant Observations. 



are not presumed beforehand to emanate from the same source 

 of error. The particular supposition concerning the a priori 

 distribution of sources which is contemplated by the De- 

 Morgan-Glaisher Method, has not perhaps been stated by- 

 its distinguished advocates. The particular assumption made 

 by the other Method is that one value of each h is as likely as 

 another over a certain range of values — not necessarily between 

 infinite limits. I have elsewhere* discussed the validity of 

 this assumption. I have also attempted to reduce the in- 

 tolerable labour involved by this method. Forming the equa- 

 tion in x of (n — 1) degrees, 



nx n ~ l — (n— lJStfj x n ~ 2 + (rc — 2)S# 1 # 2 x n ~ z — &c. = 0, 



I assume that the penultimate (or antepenultimate) limiting 

 function or derived equation will give a better value than the 

 last-derived equation \nx— \n — \ S.%, which gives the simple 

 Arithmetic Mean. Take the observations above instanced 

 under hypothesis (7), 



31, 45, 43, 100, 43, 47-5, 100. 



For convenience take as origin the Arithmetical Mean of 

 these observations 58*5, say 58. Then we have the new 

 series 



-27, -13, -15, +42, -15, -11, +42. 



Here S# 1 # 2 = — 2494. And the penultimate limiting equa- 

 tion is 



7x6x5x4x3# 2 + 5x4x3x2xlx -2494 = 0. 



Whence # 2 =119. And #=+11 nearly. To determine 

 which of these corrections we ought to adopt, the rule is to 

 take the one which makes P greatest ; which is f the one 

 which makes (x — x 1 )(x— x 2 ) (x — x z ) . . . (x—x 7 ) smallest; 

 each of the differences being taken positively. 

 The positive value, +11, gives the differences 



38, 24, 26, 21, 36, 22, 21. 

 For the negative value, —11, the differences are 

 16, 2, 4, 53, 4, 0, 53 



(where of course stands for a fraction). The continued 

 product of the second series is the smaller. Hence —11 is 



* Camb. Phil. Trans. 1885, p. 151. 



t See Phil. Mag. 1883, vol. xvi. p. 371. 



