490 LAPLACE. 



players are of equal skill and one of them has an infinite capital ; 

 there is an approximate calculation which is reproduced in pages 

 235—238 of the Theorie...des Proh. Next we have the problem 

 about balls and the long dissertation on some integrals which we 

 find reproduced in pages 287 — 298 of the Theorie...des Proh. 

 Lastly we have the theory of errors substantially coincident with so 

 much of the same theory as we find in pages 314 — 328 and 

 340—342 of the Theorie...des Proh. 



922. A theorem may be taken from page 327 of the memoir, 

 which is not reproduced in the Theorie...des Proh. 



To shew that if ^|r (x) always decreases as x increases between 

 and 1 we shall have 



I i/r [x) dx greater than S I x'^^jr (a?) dx. 

 Jo •'o 



It is sufficient to shew that 



x^ I i/r [x) dx is greater than S j x^yjr (x) dx, 



or that 2x I -v^^ (x) dx is greater than 2x^ -^ {x)y 







r X 



or that I '^ (x) dx is greater than x -v/r (x), 



"J 

 1 , r \ ' , i 1 I / \ d^ (x) 



or that Y l*^) ^^ greater than y \F) + ^ — j — ? 



but this is obviously true, for ^ is negative. 



The result stated on page 321 of the Theorie...des Proh., that 



k" . 1 . 



under a certain condition -^ is less than - , is an example of this 



theorem. 



923. In the Connaissance des Terns for 1813, which is dated 

 July 1811, there is an article by Laplace on pages 213 — 223, 

 entitled, Du milieu qiiilfaut choisir entre les residtats d'tin grand 

 nomhre d'ohservations. The article contains the matter which is 

 reproduced in pages 322 — 329 of the Theorie...des Proh. Laplace 

 speaks of his work as soon about to appear. 



