102 A. Mukhopadhyay — On Poisson's hitegral. [No. 1, 



and 4 (l + tan2 ^^ dx= / ^ (l + tan2 </>) d<p, 



in this last expression, and reducing, we get 



j_ 2 r (sin 2(/))P # 



~ (l-a2)2^-P-l J (l-2acos2<^ + a2)^ + ^-^' 

 Hence, by putting 



— cos 2<^ = cos y=— cos (tt — y), 

 2 sin 20 tZ^ = — sin 2/ <^2/> 

 2# = - dy, 

 this easily reduces to 



J _ __ 1 I sin^' y dy 



-J f_ 



(1^^2)2^-2^-1 J (l-2acos2/ + a^)^-^^~'' 

 We see, therefore, that, by the substitutions given above, the indefinite 

 integral is transformed into another in which the power of the denomi- 

 nator is depressed, provided that 



To obtain the definite integral, we have only to ascertain the new limits ; 

 and it is easy to see that, for 



as = TT, 



we have 



^2' 

 (/) = 0; 

 and for these values of </> we get 



y = o, 



y = 7r. 

 Hence, finally, we have the transformation formula 



sin^ X da; 



Jsin^ X da; _ 1 J '^ 



o (l-2a cos a7 + a2)'^~(l-a2)2^-^-l Jo (1 - 2a cos ^ + a2)l "^i'-w 

 This result holds if a ^ 1; when a y \, (1 — 0,2) must be replaced by 

 (a**— 1), as is sufficiently obvious from the above transformation. Poisson 

 characterized the above relation as a " rapport remarquable ''* ; his 

 demonstration is based on the fundamental differential equation quoted 

 above, and any one who has honestly attempted to master his proof 

 must confess it to be abstruse. 



In the particular case when p = 2n, we get 



J"^^ sin^^xdx ^ 1 r^ sin^^ x dx 



o (l-2acos^-i-a'0''"*'^ ^~"^Jo (1 - 2a cos ^r + a2)^/" ^^ 



* Loc. cit. p. 626. 



