1869.] Errors of Observations. 407 



philosophy, — one which probably never can be more than partially resolved. 

 Still, even a cursory and superficial examination of a few particular cases 

 seems to show that, far from being a mere arbitrary assumption, it is at 

 least a reasonable and probable account of what really does take place in 

 nature, in many large classes of errors of observations. The history of 

 practical astronomy, in particular, seems to prove that, whatever doubt may 

 be entertained of its exactness as applied to the errors of rude and primi- 

 tive observers, we may safely accept it in the case of the refined and deli- 

 cate observations of modern astronomers. 



It would be scarcely possible in this Abstract to convey any clear idea 

 of the mathematical analysis employed in reducing the above hypothesis to 

 calculation. It will suffice to remark that, whereas in the processes given 

 by Laplace and Poisson, when applied to the problem before us, the ele- 

 mentary component errors are at first supposed of finite magnitude, and 

 finite in number, and the results are afterwards modified for the supposition 

 that the magnitude of the errors becomes infinitesimal and their number 

 infinite ; much simplicity is gained in this Paper by making these suppo- 

 sitions at the commencement. Also, instead of taking a simultaneous view 

 of all the elementary errors, as affecting the actual or resultant error, the 

 latter is considered as produced by the superposition of some one of the 

 elementary errors upon the error produced by the combination of all the 

 others. "We are thus led to examine the infinitesimal change produced in 

 a given finite error, as expressed by a given function, by the superposition 

 of a new infinitesimal error ; and from the analytical expression arrived at, 

 it is shown how to find the form of the function of error resulting from the 

 combination of an infinite number of given infinitesimal errors. This form 

 is found to be altogether independent of the nature or laws of the compo- 

 nent errors. If we assume the following data as known, viz. 

 #2= sum of the mean values of the component errors, 

 A = sum of the mean values of the squares of component errors, 

 z'= sum of squares of the mean values of component errors, 

 it is proved that the probability of the actual resulting error being found to 

 lie between * and x + dx is 



— - 2(A-i) dx. 



This result will be found to agree with Poisson's. 



April 29, 1869, 



Lieut. -General SABINE, President in the Chair. 



Pursuant to notice given at the last Meeting, Alphonse DeCandolle, of 

 Geneva ; Charles Eugene Delaunay, of Paris ; and Louis Pasteur, of Paris, 

 were ballotted for and elected Foreign Members of the Society. 



The following communications were read ; — - 



2 h ,2 



