138 Mr MURPHY'S SECOND MEMOIR ON THE 



tlie operation A being performed on the supposition that the finite 

 increment of x is unity, and x being put =1 after the operation A""' 

 has been performed. 



Put i=l — f, and therefore, 

 A"-'(M_,r-') = A"-'M_,-^A'-'M_.(a;-l)+— — A"-'M_,(a;- 1) (a--2)-&c. 



and since m_^ is of m dimensions, the first term of this series which does 

 not vanish is 



ftn-m-\ 



- 1 ■ 2.-(«-/»- 1) •^""'"-^^'^~ ^^ (^-^)--^^ -n + m + l), 

 and therefore the whole expansion is of the form 



t'"-"-' r, 1.2.3 (w-l), 



which being substituted gives 



_ d'{t''t"'-'"-T} 



and since the equation t''t''""'-'^F'=0 has at least 2n-m—l real roots, 

 viz. ti of them =0, and n — m—l of them = 1, it follows that the w"" 

 derived equation T„ = has n — m—l real roots lying between and ] . 



COK. Since r„_. = ,.,.3.1(,_^) • ^ {r^-^u.J-^}, 

 if we actually differentiate we get 



^-^= 1.2.3.!..(«-l) -^""'^^-^-^+^>—^^ + "~^^"-^"'^- 



21. Let now <l>{x) be any function whatever, and let it be required, 

 in general, to find J'(i), so that ftj'it) ■ f = ^{x), provided x be any 

 integer from to w — 1 inclusive. 



It has been shewn in Art. 18, that a function T„.i of w— 1 di- 

 mensions may always be found to satisfy the imposed conditions, and 

 for the most general value oi f{t) we shall then have 



