312 LAGRANGE. 



573. In the problem of Art. 570 it is obvious that the sum 

 of the errors must lie between nb and nc. Hence f[z) ought 

 to vanish if z does not lie between these limits; and we can 

 easily shew that it does. 



For if z be greater than 7ic there is no term at all in f{z), 

 for every quantity raised to the power n — 1 would be negative. 



And if z be less than nh, then f{z)- vanishes by virtue of the 

 theorem in Finite Differences which shews that the n^^ difference 

 of an algebraical function of the degree n—1 is zero. 



This remark is not given by Lagrange. 



574. We will now supply what we presume would be the 

 demonstration that Lagrange must have had in view. 



Take the general problem as enunciated in Art. 570. It is 

 not difficult to see that the following process v/ould be suitable 

 for our purpose. Let a be any quantity, which for convenience 

 we may suppose greater than unity. Find the value of the ex- 

 pression 



\ i(f) {xj a^i dxjr \ Icj) {x^) a^2 dxA J j</) {x^) a^» dxA , 



where the integrations are to be taken under the following 

 limitations ; each variable is to lie between b and c, and the sum 

 of the variables between z and z + Bz. Put the result in the 



form Pa^Sz ; then I Pdz is the required probability. 



J/3 



Now to find P we proceed in an indirect way. It follows from 

 our method that 



'ne 



(f> {x) d" dx\ =i Pddz. 



J b ) J lib 



But Lagrange by a suitable transformation shews that 



' (f> {x) d'dxi = I ^f{z) a'dz, 



h ) J Zo 



where z^ and ^^ are known. Hence 



rnc fz 



Pa'dz= f[z)a'dz. 



J nb J Xo ' 



It will be remembered that a may be ani/ quantity which 



