Section III., 1885. [ lOl ] Trans. Roy. Soc. Canada. 



VIII. — On the (Icier m'walioii in lenm of a definite inlcgraJ of the value of the expression 



^{(.+«)""-.^+^->V:^- +(- 0--^T('+hO" +("')"(■■-?)■""! 



the series to he continued onJi/ as long as the quantity raised to poicer, m + n, 

 is positive, n being a positive integer, and m a positive integer, zero, or a neg- 

 ative integer namerically less than n ; and on the deduction therefrom of 

 approximate values in certain cases. By Charles Carpmael, M.A. 



(Presented May 28, 1S85.) 



Three years ago, at the first meeting of this Society, I had the honor of reading a paper 

 " On the Law of Facility of Error in the sum of n Independent Quantities, each accurate to 

 the Nearest Unit,"' which paper was printed in the first volume of our Proceedings. I 

 there made use of the approximate A-alue, found by Laplace and confirmed by Cauchy, of 

 the expression — 



y I (.+»)■-„(„.+ ■•-,)'+ ..^ +(_,)-|^(,,.+;_.)"+ ... +(-i)"(.-^)" I 



where n is any positive integer, and the series is to be continued only as long as the quan- 

 tity raised to power n is positive. 



Isaac Todhuuter, in his " History of the Theory of Probabilities," says - on the 

 Chapter iu Laplace, in which the approximate value is given, " we may observe that this 

 Chapter contains many important results, but it is to be regretted that the demonstra- 

 tions are very imperfect. The memoir of Cauchy,-' to which we have referred, is very 

 laborious and difhcitlt, so that this portion of the Théorie des Probabilités remains in an itn- 

 satisfactory state." 



This remark led to the investigations that I now propose to lay before you, which 

 (while they confirm the resitlts of Laplace and Cauchy in the case which ^vas quoted in 

 m}- previous paper, and also, if we correct some numerical errors in Cairchy's results, in 

 other similar cases) shew that a greater objection than that of being "very laborious and 

 difficult" may be taken to Cauchy's proof, namely, that a large nitmber of his intermediate 

 results are erroneous in the jiartictxlar cases in which he employs them. The results to 

 whic-h I refer are the values which he obtains for certain extraordinary integrals, to which 

 attention will be drawn in the sequel. 



Let us for brevity denote by (p (x, m, //) the expression — 



,À.[(:'+'i) -«('+f->) - +(-')Ftnr(-'+l-') - +(-0 (-?) \ 



' In the paper as printed, the word " degree" was, by printers' error, substituted for " unit." 



' History of the Tlieory of Probabilities, § 968, p. 526. 



■■ The memoir of Cauchy is publi.«hed in the Journal de l'École Polytechnique, 28e cahier. 



