STORY. — A NEW GICNERAL THEORY OF ERRORS. 199 



XIII. The Ordinary Case. 

 In the ordiuary theory it is assumed that 

 (108) Pk^O for 3^1; 



so that the only P regarded as uot negligible is P^ =: 1, which being 

 substituted in (77) with the approximate value C'o = — : , by (101), 



gives, by (51) and (63), 



(s) =CoP,S, ^ - 



VStt 



$ (^s) = has, then, no real finite root, so that 



s = — cc , s ^ -\- cc 



and (80) gives again the same value of Co as (101), which is, therefore, 

 the most exact value possible on the assumption made. We have fur- 

 ther, by (78) and (27), 



(109) f{s) = -^ e"^ .A(^) = —^ e-^; 



while (GO) gives 



1 p —1 



Pk = —= / s* • e 2 . ds, 



— 00 



that is 



(110) P^n,+i = for O^m 

 and 



(111) p2m =r 2"* (m - I)""' or p.,„ = 2™ (m — i)""' • ^^-n 



for ^ m, so that, far from being negligible for the greater values of 

 m, p2m increases beyond all limit as m increases. 



XIV. Two Other Particular Cases. 



For the purpose of illustrating the variety of forms the general theory 

 may take in particular cases, and especially its possible deviation from 



the ordinary theory, we have plotted the curve represented by y = -f{s) 



(the probability-curve) for a = 0, /3 = 1 and for a := 2, /3 = 0, using the 

 approximate values of Cq, C^, and C^ given by (101), for such different 

 values of pg and p^ (in the two cases, respectively) as to show all the 



