560 LAPLACE. 



»' —00 '' —00 



Transforming as before we obtain 



e-"' (cos 10+ sT^l sin 1 6) sin^ 6 r^^+^ Jr ^6^. 



•'0 '' 



Now sin^^ may be expressed in terms of sines or of cosines 

 of multiples of 6, according as q is odd or even, and the highest 

 multiple of 6 will be qO. And we know that if m and n are 

 unequal integers we have 



/•27r 



I sin mO cos nO dO — 0, 





 r2rT 



I. 



cos mO cos n6d6 = 0, 







2n 



sin md sin nd dd = ; 



thus the required result is obtained. 



Laplace finally takes the same problem as Daniel Bernoulli 

 had formerly given ; see Art. 420. Laplace forms the differential 

 equations, supposing any number of vessels ; and he gives without 

 demonstration the solutions of these differential equations : the 

 demonstration may be readily obtained by the modern method 

 of separating the symbols of operation and quantity. 



1001. Laplace's Chapter IV. is entitled, De la prohahiUte des 

 erreurs des resultats nioyens d\in grand nomhre d observations, et 

 des resultats moyens les plus avantageux : this Chapter occupies 

 pages 304—348. 



This Chapter is the most important in Laplace's work, and 

 perhaps the most difficult ; it contains the remarkable theory 

 which is called the method of least squares. Laplace had at an 

 early period turned his attention to the subject of the mean to be 

 taken of the results of observations ; but the contents of the pre- 

 sent Chapter occur only in his later memoirs. See Arts. 874, 892, 

 904, 917, 921. 



Laplace's processes in this Chapter are very peculiar, and it is 

 scarcely possible to understand them or feel any confidence in 



