LAPLACE. 569 



100-i. We take Laplace's next case. 



Let ry^ = 7^ = • • • = 7s = 1- ^^^ ^^^^ limits of error he ±a; sup- 

 pose that the function of the facility of error is the same at every 

 observation, and that positive and negative errors are equally 

 likely : thus / (— x) =f{x). 



Here h = 0, A^=|^', ^=0, K' = ~h'. 



By equation (4) of Article 1002 the probability that the sum 

 of the errors at the s observations will lie between — rj and 77 

 is 



dv. 



2 [\-2sk' 



This will be found to agree with Laplace's page 808. 



'a r a 



We have k' = z''f{z)dz=2 z'f{z) dz, 



a ra 



and 1 = f[z) dz=2 f{z) dz ; 



-a 



hence i^fiz) always decreases as z increases from to a we see, as 



in Art. 922, that h' is less than -^ . 



o 



1005. Laplace next considers the probability that the sum of 

 the errors in a large number of observations will lie between 

 certain limits, the sign of the error being disregarded, that is all 

 errors being treated as positive ; the function of the facility of 

 error is suj)posed to be the same at every observation. 



Since all errors are treated as positive, we in fact take nega- 

 tive errors to be impossible ; we must therefore jDut 5 = in 

 Poisson's problem. 



Take 7^ = ^3 = • ■ • = 78 = 1. Then 



Take c = Z; then, by equation (-i) of Art. 1002, the probability 

 that the sum of the errors will lie between l—r] and l-\-7] is 



