Discordant Observations. 373 



just referred to — that the limit must be pushed forward as 

 much again ; so that the suspected observation falls within 

 the corrected limit. I have similarly treated the example 

 given by Prof. Merriman in his Textbook on The Method of 

 Least Squares (131). The limit found by him is 4'30, and 

 he therefore rejects the observation 4*61. But I find that 

 this observation is well within the corrected limit *. 



II. (3) Prof. Peirce's criterion is open to the same objec- 

 tions as that of Prof. Chauvenet. Indeed it presents additional 

 difficulties. If by y the author designates that quantity which 



Prof. Stone calls -, and which I have termed the " a priori " 



probability of a mistake, I am unable to follow the reasoning 

 by which he obtains a definite value for this y. But I am 

 aware how easy it is on such subjects to misunderstand an 

 original writer. 



III. We come now to the third class of method, of which 

 I am acquainted with three species. (1) There is the procedure 

 indicated by De Morgan and developed f by Mr. Glaisher ; 

 which consists in approximating to the weights which are to be 

 assigned to the observations respectively, after the analogy of 

 the Reversion of Series and similar processes. (2) Another 

 method, due to Prof. Stone J, is to put 



P = M 2 • • • e-^-^-W*-*^ ' * * xdh x dh 2 . . . 

 as the a posteriori probability of the given observations having 

 resulted from a particular system of weights h x 2 h 2 %1 &c, and a 

 particular Mean x ; and to determine that system so that P 

 should be a maximum. (3) Another variety is due to Prof. 

 Newcomb§. 



III. (1) & (2) Neither of the first two Methods are well 

 adapted to the first two hypotheses. Both indeed may success- 

 fully treat mistakes by weighting them so lightly as virtually to 

 reject them. But both, I venture to think, are liable to err in 

 underweighting observations, which, upon the first two hypo- 

 theses, have the same laic of frequency as the others. Both, in 

 fact, are avowedly adapted to the case wdiere the observations 



* These corrections may be compensated by another correction to which 

 the method is open. In determining whether the suspected observation 

 belongs to the same type as the others, would it not be more correct to 

 deduce the characters of that type from those others, exclusive of the 

 suspected observation ? The effect both on the Mean and the Modulus 

 would be such as to contract the limit. 



t Memoirs of the Astronomical Society. 



\ Monthly Notices of the Astronomical Society, 1874. This Method 

 was proposed by the present writer in this Journal, 1883 (vol. xvi. 

 p. 360), in ignorance of Prof. Stone's priority. 



§ American Journal of Mathematics, vol. viii. No. 4. 



