LAGRANGE. 309 



This result expresses the probability that the error in the 



mean result will lie between — and — on the followinof hy- 



n n o J 



pothesis ; at every trial the error may have any value between 

 — h and + h ; positive and negative errors are equally likely ; 

 the probability of a positive error z is proportional to h — z, and 



in fact ~ — — - is the probability that the error will lie be- 

 tween z and z + Sz. 



We have followed Lagrange's guidance, and our result agrees 

 with his, except that he takes 7i = l, and his formula involves 

 many misprints or errors. 



568. The conclusion in the preceding Article is striking. We 

 have an exact expression for the probability that the error in 

 the mean result will lie between assigned limits, on a very 7^ea- 

 sonahle hypothesis as to the occiirrence of single errors. 



Suppose that positive errors are denoted by abscissse measured 

 to the right of a fixed point, and negative errors by abscissae 

 measured to the left of that fixed point. Let ordinates be drawn 

 representing the probabilities of the errors denoted by the re- 

 spective abscissae. The curve which can thus be formed is called 

 the curve of errors by Lagrange ; and as he observes, the curve 

 becomes an isosceles triangle in the case which we have just 

 discussed. 



569. The matter which we have noticed in Arts. 563, dQ>^, 

 566, 567, 568, had all been published by Thomas Simpson, in his 

 Miscellaneous Tracts, 1757 ; he gave als(3 some numerical illus- 

 trations : see Art. 371. 



570. The remainder of Lagrange's memoir is very curious ; 

 it is devoted to the solution and exemplification of one general 

 problem. In Art. 567 we have obtained a result for a case in 

 which the error at a single trial may have any value between 

 fixed limits ; but this result was not obtained directly : we started 

 with the supposition that the error at a single trial must be one 

 of a certain specified number of errors. In other words we started 

 with the hypothesis of errors changing per saltum and passed on 



