INVERSE METHOD OF DEFINITE INTEGRALS. 1^9 



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Now 7'„-i contains only n constants, being of » — 1 dimensions, and 

 therefore if we denote by P„ the same quantity as in the preceding 

 Section, namely the coefficient of h" in 



{1- 2h{l-2t) + h:'}-i, 



we may put 



T„.i = ttoPo + a^P, + (hP2 + + a„_,P„_i, 



the right-hand member being of the same dimensions with the left, and 

 containing the same number of constants. 



Now by the properties of P„ we have j;P„P„ = 0, when m and « are 

 unequal, and 



2« + l 

 Hence we have fiP^T„_.,= «„ 





Hi 



Jl'* 2 -* n-1 — "^ • 



But by the conditions of the question, 



jc being any integer less than n. 



Hence 

 j;P„7;-i = ^r„_i = 0(O) = (f>{h) when h is put =0, 



iP 7'„_,=j;2;_, (1 - 2o=0(o)-20 (1)= - A ^^y^ .<^ (A), 



