LAGRANGE. 311 



Let c-'h = t, and expand {a' — a^y by the Binomial Theorem ; 

 thus ' W (fi {x) a" dx\ 



b 



^j— yja -na + 2. 2 ~***l/ ^ «^ «^- 



Now decompose I ^" ^oP'dy into its elements; and multiply 





 them by the series within brackets. AVe obtain for the coefficient 



of a^'^ the expression 



where the series within brackets is to continue only so long as the 

 quantities raised to the power n — 1 are positive. 



Let nc—y = z ; then dy — — dz\ when y—^ we have z = nc, 

 and when y = 00 we have = — 00 . Substitute nc — z for ?/, and 

 we obtain finally 





where f{z) = \ [nc — zy ^ - n {nc - z -ty 



+ \,2 ' {nc-z-2ty'-.. 



the series within brackets being continued only so long as the 

 quantities raised to the power n — 1 are positive. 



Lagrange then says that the probability that the sum of the 

 errors in n observations will lie between /3 and 7 is 



f V(.) dz. 



J/3 



572. The result is correct, for it can be obtained in another 

 way. We have only to carry on the investigation of the problem 

 enunciated in Art. 563 in the same way as the problem enunciated 

 in Art. 564 was treated in Art. 567; the result will be very similar 

 to those in Art. 567. Lagrange thus shews that his process is 

 verified in this example. 



