308 LAGRANGE. 



the series tuithin the brackets is to continue only so long as r is 

 positive in yjr (r). We will denote this by F^y). 



The probability that the mean error will lie between /3 and y, 

 where 7 is greater than /3, is F[y) — F {^) if we include 7 and 

 exclude ^ ; it is F{y — 1) — F{/3 — 1) if we exclude 7 and include 

 /9; it is F{y)—F[(3—1) if we include both 7 and yS ; it is 

 F{y — 1) —F{/3) if we exclude both 7 and /3. 



It is the last of these four results which Lagrange gives. 



We have deviated slightly from his method in this Article in 

 order to obtain the result with more clearness. Our result is 

 F {y — 1) — F {^) ; and the number of terms in F (y-l) is de- 

 termined by the law that r in ^fr (r) is always to be positive : 

 the number of terms in F ((3) is to be determined in a similar 

 manner, so that the number of terms in F (/3) is not necessarily 

 so great as the number of terms in F (y — l). Lagrange gives an 

 incorrect law on this point. He determines the number of terms 

 in F{y — 1) correctly; and then he j^'^ohngs F {,6) until it has 

 as many terms as F{y — 1) by adding fictitious terms. 



567. Let us now modify the suppositions at the beginning 

 of Art. 564^. Suj^pose that instead of the errors — a, — (a — l), ... 

 we are liable to the errors — ka, — k {a — l), ... Then the investi- 

 gation in Art. 5Q4! gives the probability that the error in the mean 



ak 

 result shall be equal to — ; and the investigation in Art. 5G6 



gives the probability that the error in the mean result shall lie 



between — and - . Let a increase indefinitely and k diminish 

 71 n 



indefinitely, and let o.k remain finite and equal to h. Let 7 and /3 



also increase indefinitely ; and let 7 = c« and I3 = h(x where c and h 



are finite. We find in the limit that F [y] — F {/S) becomes 



J- L + ny''-2n (c + n-ir^ + ^"^ ^^"^ ~ ^^ (c + n- 2^- .. 



each series is to continue only so long as the quantities which 

 are raised to the power 2n are positive. 



