LAGRANGE. 813 



is greater than unity. We shall shew that we must then have 



Suppose that z^ is less than nh^ and z^ greater than nc. Then 

 we have 



1 f{z)a'dz+ {f{z)-F} a'dz'-\- f{z)a'dz = ^, 



J SSq J nb '1 nc 



for all values of a. Decompose each integTal into elements ; put 

 a^^ = p. We have then idtimately a result of the following 

 form 



a- 



^0 jr, + T^p + V + ^3P'+ - ^^^ ^V- •••} = 0, 



where T^, T^,... are independent of p. And p may have any 

 positive value we please. Hence by the ordinary method of in- 

 determinate coefficients we conclude that 



Thus P=f{^)- 



The demonstration will remain the same whatever supposition 

 be made as to the order of magnitude of the limits z^ and z^ 

 compared with nh and 7ic. 



57o. Lagrange takes for another example that which we have 

 akeady discussed in Art. 567, and he thus again verifies his 

 new method by its agreement with the former. 



He then takes two new examples ; in one he supposes that 



<f) {x) = K \/ c^ — x\ the errors lying between — c and c; in the 

 other he supposes that cj) (x) = Kcosoo, the errors lying between 



- :=: and ^ . 

 2 2 



576. We have now to notice another memoir by Lagrange 

 which is entitled Becherches sw les suites reciirrentes dont les 

 termes varient de plusieurs manieres di0rentes, qu sur Vintegra- 

 tion des equations lineaires aux differences jinies et partielles ; et 

 sur Tusage de ces equations dans la theorie des hazards. 



This memoir is published in the Nouveaux Menioires de VAcad. 

 ... Berlin. The volume is for the year 1775; the date of pub- 



