568 LAPLACE. 



Therefore (2) becomes 



P=— \ e cosilx — ex) sm rix — 



TTJo ^ 



2L r^ -/c2^2 . \ 2 • 7 



-\ \ e sm [Ix — ex) x^ sm 7)X ax. 



TT J 



We formerly transformed the first term in this expression of P; 

 it is sufficient to observe that the second term may be derived 

 from the first by differentiating three times with respect to / and 

 multiplying by l^ ; so that a transformation may be obtained for 

 the second term similar to that for the first term. 



1003. Laplace gives separately various cases of the general 

 result contained in the preceding Article. We will now take his 

 first case. 



Let 7i = 72= ••• =78= 1- Suppose that the function of the 

 facility of error is the same at every observation, and is a constant ; 

 and let the limits of error be + a. Then 



J —a 



If C denote the constant value of f{z) we have then 



2a (7=1. 



Here Jc = 0, h' = -^— = — , 7^^ = 77 , 



Letc = 0; then by equation (4) of the preceding Article the 

 probability that the sum of the errors at the 5 observations will 

 lie between — 77 and 77 



2a^[s'7r) J -ri ^VM Jo 



Let —^^ = f; then the probability that the sum of the errors 

 will lie between — ra ^s and ra V^ 



This will be found to agree with Laplace's page 805. 



