LAGRANGE. 305 



are not known d priori; but that ct, /S, y, ... are known by ob- 

 servation. Then in the sixth problem it is taken as evident that 

 the most probable values of a, h, c, ... are to be determined from 

 the results of observation by the relations 



a = ;8 = 7"--" 

 so that the value of - of the jjreceding Article may be written 



fM _ pa. + ql3 +ry+ ... 

 n a + /3 + 7+... 



Lagrange proposes further to estimate the probability that the 

 values of a, h, c, ... thus determined from observation do not differ 

 from the true values by more than assigned quantities. This is an 

 investigation of a different character from the others in the 

 memoir; it belongs to what is usually called the theory of in- 

 verse probability, and is a difficult problem. 



Lagrange finds the analytical difficulties too great to be over- 

 come ; and he is obliged to be content with a rude approxi- 

 mation. 



563. The seventh problem is as follows. In an observation it 

 is equally probable that the error should be any one of the 

 following quantities —a, - (a - 1), ... — 1, 0, 1, 2 ... 13 ; required 

 the probability that the error of the mean of n observations shall 

 have an assigned value, and also the probability that it shall lie 

 between assigned limits. 



We need not delay on this problem ; it really is coincident 

 with that in De Moivre as continued by Thomas Simpson : see 

 Arts. 14^8 and S64<. It leads to algebraical work of the same kind 

 as the eighth problem which we will now notice. 



oQ4^. Suppose that at each observation the error must be 

 one of the following quantities — &, - (a — 1), ... 0, 1, ... a ; and 

 that the chances of these errors are proportional respectively to 

 1, 2, ... a + 1, a, ... 2, 1 : required the probability that the error in 



Lb 



the mean of n observations shall be equal to - . 



20 



