1888.] A. Mukhopadhyay— Ow Poisson's Integral. 103 



As the power of the denominator on the right hand side is by one less 

 than that on the sinister, this obviously serves as a formula of reduction, 

 and it at once follows from (4) that 



sin^^ X dx TT (2n) ! 



X 



_ ... (7) 



(l-2acosa;-+-a2)^ + ^ 2^''{i-a^) [nl}^ 



o 



This formula may be regarded as supplementary to equation (38) in 

 Bertrand's Galcul.,t.ll, 153. Putting 7i = in (6), we have the well- 

 known result 



£ 



^^ "" (8) 



Jq 1 — 2a cos j7 + a2 1 — a^ 

 which might also have been obtained by putting ti = in (7) 

 Again, if we substitute 7i = in (5), we have 



Jsin^ X dx 1 I . -n 7 

 r = I sin-^ £C ax. 

 o (l-2acos«;+a8)^ + l (l-^^^-^Vo 

 The value of the right hand side depends on the form of p. li p =z 2r, 

 we get 



X^ sin^^ X dx _ TT (2r) ! 



(1- 2a cos a;+a2)2^+l~22''(l-a2)2^ + l ir iV 



If p = 2r+l, we get 



"" 2r + l ^^^ o2r+l {r \] 



(9) 



X 



sin^^'-^^xdx 22^+1 V i ...(10) 



o (l-2acos^-ha2)2(-+l) (l-a^j^C^ + l) (2^+1)' 



§. 3. Symbolic value for TT. 



We shall now give a symbolic value for it, to which we are easily 

 led by the integral 



'""" X sin £C dx rr^ 



i: 



'o l + cos^o? 4 



which is also considered by Poisson.* Poisson has effected the evalua- 

 tion of this by expanding, and integrating each term separately ; the 

 substance of his process is well reproduced by Bertrand (t. II, 150). 

 It is easy to see that this may also be integrated by parts, since 



r"^ X sin x dx C^ i ( - 1 1 



Jo i + cos2^ " " Jo ^ l*^''' "''' ''^ 



= — J ic tan (cos x)\ -f / tan (cos x) dx =z — 

 ^ ^ x=0 Jo * 



* Loc. cit. p. 623. 



