512 LAPLACE. 



ferential equations; he concludes that such functions are ad- 

 missible under certain conditions. Professor Boole reofards the 

 argument as unsound ; see his Finite Differences, Chapter x. 



955. Laplace closes the Chapter with some general considera- 

 tions respecting generating functions. The only point to which we 

 need draw attention is that there is an important error in page 82 ; 

 Laplace gives an incomplete form as the solution of an equation in 

 Finite Differences ; the complete form will be found on page 5 of 

 the fourth supplement. We shall see the influence of the error 

 hereafter in Arts. 974, 980, 984. 



956. We now arrive at the second part of Livre i., this is 

 nearly a reprint of the memoir for 1782; the method of approxi- 

 mation had however been already given in the memoir for 1778. 

 See Arts. 894, 899, 907, 921. 



The first chapter of the second part of Livre I. is entitled Be 

 Tintegixttion jjar approximation, des differe^itielles qui renferment 

 des facteurs Sieves a de grandes puissances; this Chapter occupies 

 pages 88—109. 



957. The method of approximation which Laplace gives is of 

 great value : we will explain it. Suppose we require the value of 



\ydx taken between two values of x which include a value for 



which y is a maximum. Assume y — Fe-^^, where F denotes this 

 maximum value of y. Then 



jydx=YJe-<^^dt. 



Let y = (j) {x) ; suppose a the value of x which makes y have 

 the value Y : assume x= a+ 6. 



Thus (^(a + (9)= Ye-''; 



Y 



therefore f = log -7-7 ^r . 



(p {a-^ o) 



From this equation we may expand ^ in a series of ascending 

 powers of 6, and then by reversion of series we may obtain 6 in a 

 series of ascending powers of t. Suppose that thus we have 



