( 226 ) 

 Mathematics. Extract of a letter of Mr. V. Williot, to the Academy. 



Ill Ills s})lendi{l work entitled : "Theorie, propriétés, formules de 

 transformation et methodes d'évahiation des integrates définies" 

 Mr. BiKRKNS DK Haan takes as basis to determine the general formnlae 

 (143, 144, 145, 146) of page 134 a definite discontinnons integral 

 the value of A^hicli has l)een established farUier on in the work 

 (Partie IIT, Methode J), N". J6) at page 333 as 



CO 



ƒ 



(hi; Jt , 

 sin {p,v) cos {(i.v) — = — for p ■= q 

 ,r, 4 



(1) 



for p <^ q 



the value willi res[)ecl to the discontiiuulv /> ^ (/ beiiig Ihe mean 

 of the extreme values. 



But he gives this result on |)age 133 in the forui: 



CD 



(b; jt , 



,v cos qx — = — Jor q <^ r 



w 2 

 for q > r 



SO that in the continuation of his deduction we lind (hat the term 

 corresponding to 7 = r amounts to doubk» the value of the real 

 value and that the genei-al formulae of page 134 are 1o be rectified 

 in this Avay as well as the ai)i>lications. 



Particularly on page 639 foruiula 1900 we liud 



J 



sin X cos a.v n » 



d.v — — ^ 7>" 



l — 2pcos.v-^p>^ X 2p n 



:tp 



a— 1 



whilst the exact value of this integral is 



V 



a—\ 



_l_ ^,. _!_ ^,.+1 + y/'+2 _!_ 



2 1-; 



4^ 



\-p 



And really writing al'ter multi])lication by p 

 p sin X cos ax 



J 



1 — 2 p cos X -\-p'^ 



dx 



4^ 1— p 



it is sufficient to develop the first factor of the function of which 

 the inteo-ral is to be found 



p sin X 



rrr S p''' sin kx 



l — 2p cos X -]- p* /,•=! 



to refind by means of the integral (1) the development of the second 

 term of the equation: 



