124 Mr MURPHY'S SECOND MEMOIR ON THE 



because the u under the logarithmic sign is the same as if we had 



placed there, a or any arbitrary symbol, and is therefore treated as a 



(jLu 

 constant in the differentiation; hence the coefficient of h" in ~t- is the 



term independant of m in 



(> - -:) 



n + i ' 



that is, its value is Q„, and therefore by the preceding Article /Q.r 

 vanishes between the limits of x, and n — \, its general value being 



T being the same function of / that U is of u. 



By this theorem, every possible variety of rational and entire func- 

 tions which possess the above-mentioned property may be found, as in 

 the following 



Example: 



To find a rational function of t, in which the powers of the variable 

 are in arithmetical progression, such that jiQ,nt'=0 when x is any number 

 of the series 0, 1, 2 {n — 1). 



In this instance put U = 1 — u"", m being any positive integer. 

 Hence Q„ = term independent of u in 



/ t\ "*""*"'* 



(i-^o».(i--) 



^ « (w + l)(w + 2)...(w+m) w. (w-l) (M+l)(w+2)...(?i+2OT) ^,„_. 



1' 1.2...m ■ 1.2 ■ 1.2, ..2m 



in which if we take in particular m =1, we get the value of P„ before 

 found in Art. (2). 



This formula for Q„ may be written in another form by which it 

 will comprise the case where /w is a fraction, thus 



n (m+l)(m+2)...( m+n) ,„, w(«+l) (2»^+l)(2w^+2)...(2w^-^w) „ „ 



^-=^-i- r^:::^ -^ ^"ttt- t:2::ji -^ "*'''• 



