LAPLACE. 589 



The titles of some other memoirs on the subject of least squares 

 will be found at the end of the Treatise on Probability in the 

 Encyclopcedia Britamiica ; we would also refer the student to the 

 work by the Astronomer Eoyal On the A Igebraical and Numerical 

 Theory of Errors of Observations and the combination of Observa- 

 tions. 



1018. Laplace's fifth Chapter is entitled Application du Calcid 

 des Probabilites, a la recherche des phenomenes et de leurs caicses : 

 it occupies pages 31;9 — 3G2. 



The example with which Laplace commences will give a good 

 idea of the object of this Chapter. Suppose that observations 

 were made on 400 days throughout which the height of the 

 barometer did not vary 4 millimetres ; and that the sum of the 

 heights at nine in the morning exceeded the sum of the heights 

 at four in the afternoon by 400 millimetres, giving an average 

 excess of one millimetre for each day : required to estimate the 

 probability that this excess is due to a constant cause. 



We must examine what is the probability of the result on 

 the supposition that it is not due to any constant cause, but 

 arises from accidental perturbations and from errors of ob- 

 servation. 



By the method of Art. 1004, supposing that it is equally pro- 

 bable that the daily algebraical excess of the morning result over 

 the afternoon result will be positive or negative, the probability 

 that the sum of s excesses will exceed the positive quantity c 



,00 V 



i2 



2sk' ^^ 



^^{2k's7^) J, 



= —r- I e~^' dt, where t = ,,^ ,,, . 

 VttJt f^{zsk) 



Hence the probability that the sum will be algebraically less 

 than c is 



1 r 



