LAGRANGE. " 307 



Art. 564>, we may if we please put — //, instead of /jl, without 

 changing the vakie obtained. It is obvious that this would be 

 a gain in practical examples as it would diminish the number 

 of terms to be calculated. 



This remark is not given by Lagrange. 



566. We can now find the probability that the error in the 

 mean result shall lie between assigned limits. Let us find the 

 probability that the error in the mean result shall lie between 



and - , both inclusive. We have then to substitute in the 



n n 



expression of Article 56^ for //, in succes.sion the numbers 



— ny, —(iia — l), ...7 — 1, 7, 



and add the results. Thus we shall find that, usingf ^, as is 

 customary, to denote a summation, we have 



S^ (7ia + /^ + 1) = — -^/r (7ia + 7 + 1), 



where -^/r (?-) stands for 



r (r + 1) (?' + 2) . . . (r + %i - 1). 



When we proceed to sum <^ (?ia + //- — ct) we must remember 

 that we have only to include the terms for which noL + fi — a is 

 positive; thus we find 



Xcj) (na + fJ' — OL) = ~ yjr {iiOL 4- 7 - a). 

 Proceeding in this way we find that the probability that the 



710L "V 



error- in the mean result will lie between and - , both in- 



n n 



elusive, is 



r^r- J-v/r (nOL + 7 + 1) — 2/1 -v/r (nct. + 7 + 1— a— 1) 



+ \ ^ — ' -^ (71a + 7 + 1 - 2a - 2) 



2w i^n - 1) (2?i - 2) , , , , . „ ^s . \ 

 ^ ^ ./ ^^ ^ ^ {no, + 7 + 1 - 3a - 3) + . . .| ; 



20—2 



