o 



10 LAGKANGE. 



to the supposition of continuous errors. Lagrange wishes to solve 

 questions relative to continuous errors without starting with the 

 supposition of errors changing per saltum. 



Suppose that at every observation the error must lie between h 

 and c; let ^ {x) dx denote the probabiUty that the error will lie 

 between x and x-\-dx\ required the probability that in n obser- 

 vations the sum of the errors will lie between assigned limits say 

 /3 and 7. Now what Lagrange effects is the following. He trans- 

 forms \\ ^{pc)a^dx\ into \f{z)(idz, where f{z) is a known 



function of z which does not involve a, and the limits of the 

 integral are known. When we say that f {£) and the limits of 

 z are known we mean that they are determined from the known 

 function ^ and the known limits h and c. Lagrange then says 

 that the probability that the sum of the errors will lie between 



/? and 7 is 1 f{z) dz. He apparently concludes that his readers 



will admit this at once ; he certainly does not demonstrate it. 

 We will indicate presently the method in which it seems the de- 

 monstration must be put. 



571. After this general statement we will give Lagrange's 

 first example. 



Suppose that ^ {x) is constant = K say ; then 



6 (x) «* dx = — i^ , 



J h loof a 



b\n 



therefore \j (b(x)a'dx[ = — —^ — r-f— 



Vb^' j (log a)" 



Now we may suppose that a is greater than unity, and then it 

 may be easily shewn that 



J 



n-l 



(log ay' 



thus \ (f){x) a^ dx\ = -^ {a' - a^ j if ''a'" dy. 



