INVERSE METHOD OF DEFINITE INTEGRALS. 137 



«_, = (- 1)"+'. 1.2.3....n/-~-.u.2, 



&c. = &c. 



from whence the values of A^, Ai, A-z, &c. are known, and being sub- 

 stituted, give 



J .M_2^ 



n.(«.H)(.-f2) («-l)(.-2) I 



^ 1.2 1.2 ' J 



Example : 

 Let 0(a;) = 1, then «, = 1, and therefore 



«.(« + l)(w + 2) (w-l)(/?-2) 

 "*" 1.2 ■ 1.2 



.f-&e.| 



20. The function Tn-\ possesses a property analogous to the charac- 

 teristic property of those in the former Section, that is, the equation 

 2\_^ = admits of n — m-l roofs between and 1, and consequently 

 vanishes an indefinitely great number of times between the limits / = 

 and t=\ when n is taken indefinitely great. 



For since r„., = (- 1)-' |«M., - ^^^^jtil . ^V «., ^ 

 n{n+l){n+2) {n-l)(n-2) ] 



■^ TTa • 1.2 •«-3^&c.j 



_ (-1)- ^\t^(u -VlzI u 1 1 (»-i)-(^^-a) „ t. .,„^l 

 = i.2.3....(.*-i)-rfrr \ ' 1 ^ ^'^^ i:% -"-a^-^c.JI 



= -r 2.3..U-i) -£^^'^"'"-^^"'^- 



