478 LAPLACE. 



and put for a in succession and 1 ; then we have to compare - with 



71 



2 . v . n , X 



^ ; that is, we have to compare t- with — . Now consider — ; the 



differential coefficient with respect to x is ^^ — ; so that -^ 



increases as x changes from to , — - , and then diminishes. 



* log 2 ' 



Laplace treats the same question in the Theorie...des Proh. 

 page 406 ; there also the difficulty is dismissed with the words il 

 est facile de voir. In the memoir prefixed to the fourth volume of 

 Bowditch's Translation of the Mecanique Celeste, page 62, we read : 



Dr Bowditch himself was accustomed to remark, " Whenever I meet 

 in La Place with the words ' Thus it plainly appears' I am sure that 

 hours, and perhaps days of hard study will alone enable me to discover 

 how it plainly appears." 



892. The pages 240 — 258 of the memoir contain the im- 

 portant but difficult investigation which is reproduced in the 

 Theorie...des Proh. pages 262 — 272. Laplace gives in the memoir 

 a reference to those investigations by Lagrange which we have 

 noticed in Art. 570 ; the reference however is omitted in the 

 Theorie...des Proh. 



893. Laplace now proceeds to the subject which he had con- 

 sidered in a former memoir, namely, the probability of causes as 

 deduced from events; see Art. 868. Laplace repeats the general 

 principle which he had already enunciated in his former memoir; 

 see Art. 869. He then takes the problem which we have noticed 

 in Art. 870, enunciating it however with respect to the births of 

 boys and girls, instead of the drawings of white and black balls. 

 See Art. 770. 



894. Laplace is now led to consider the approximate evalu- 

 ation of definite integrals, and he gives the method which is repro- 

 duced almost identically in pages 88 — 90 of the Theorie.,.des Proh. 



He applies it to the example x^(l—xydx, and thus demon- 

 strates the theorem he had already given ; see Art. 871 : the pre- 

 sent demonstration is much superior to the former. 



