526 LAPLACE. 



Hence he deduces 



/• 00 



I x^^ e-^ (e-^ - 1)" dx 



^n - ^ -_o 



6-* 



:&-^ e~^ dx 



Laplace calculates tlie approximate value of this expression, 

 supposing { very large. He assumes that the result which he 

 obtains will hold when the sign of { is changed ; so that he obtains 

 an approximate expression for AV; see page 159 of his work. 

 He gives a demonstration in the additions; see page 474 of the 

 Tkeorie...des Proh. The demonstration involves much use of the 

 symbol \/(- !)• Cauchy gives a demonstration on page 247 of the 

 memoir cited in Art. 964. Laplace gives another formula for 

 AV on his page 163 ; he arrives at it by the aid of integrals with 

 imaginary limits, and then confirms his result by a demon- 

 stration. 



967. Laplace says, on his page 165, that in the theory of 

 chances we often require to consider in the expression for AV only 

 those terms in which the quantity raised to the power t is positive; 

 and accordingly he proceeds to give suitable approximate formulae 

 for such cases. Then he passes on to consider specially the ap- 

 proximate value of the" expression 



Tl [71 — 1 ) 



(n-\-r sJnY - n (ii + r V?i - 2)^ + (n + r ^Jn- iy - ... , 



where the series is to extend only so long as the quantities raised 

 to the power yu, are positive, and />t is an integer a little greater or 

 a little less than n. See Arts. 916, 917. 



The methods are of the kind already noticed ; that is they are 

 not demonstrative, but rest on a free use of the symbol a/ (— 1). 



A point should be noticed with respect to Laplace's page 171. 

 He has to establish a certain formula; but the whole difficulty of 

 the process is passed over with the words determinant convenable- 

 ment la constante arhitraire. Laplace's formula is established by 

 Cauchy ; see page 240 of the memoir cited in Art. 964. 



968. In conclusion we may observe that this Chapter contains 

 many important results, but it is to be regretted that the demon- 



