INVERSE METHOD OF DEFINITE INTEGRALS. 325 



9. But whatever may be the value of m, the quantities Q„, q„ may 

 always be simply expressed in terms of t by the theorem of T^eibnitz, 

 viz. 



d"(uv)_ cV'v clu d'"^v 7i.{n — l) dHi^ f/"'^ v 



after {yiplying which we may substitute for t its value in terms of (p. 

 Thus when m= — ^ 





1.3.5....(2«-1) n 2n-l 



2.4.6....2W ^' ~1- 1 " 



• ' J. n{n-l) {2n-l){2n-3) , ,„_, _ „ . 



_ 1 .3.5.. ..{2n - 1) , 2n{2n-l) 



-" 2.4.6... .2» ^^ 1.2 " 



, 2^(2?^-l)(2>^-2)(2«-3) ,,^,„., . 



"^ 1.2.3.4 ^^ "^''•^ 



•^ 2.4.D....2ra ^ ' ' ^ 



and in the same way we have 



d''(tt'Y*i 



^"~ rr2. 3. ...«<//" 



_ 3.5.7....(2» + l) » 2;» + l 



~ 2.4.6....2« ^^^ 1- 3 ^' " 



w(w-l) (2w + l)(2«-l) 5, 3 „ , 



= I 3.5.7.--.(2?? + l) , ^--- . ^ , — i /.h2»+2) 



2V"=l"2.4.6....(2« + 2) ^^^ +^^^) -(^ _V-1#0 \, 



uu2 



