364 Mr MURPHY'8 THIRD MEMOIR ON THE 



pT'' = Uo- U,k + U,¥ - U^¥ + he. 



— f-f^§--} 



III. Since f =\ +x\i.\.t+ -^ . (h. 1. tf + j-^-j ■ (h. 1. <)' + ««:■ 



by means therefore of a single integral, x" may be adapted to general 

 differentiation. 



As this result is remarkable, we may confirm it by the general 

 rule in the First Memoir. (Vide Sect, i.) Thus, 



1 



put (x) = :j T— =l-\-xk + a^J^ + &c. 



*±i 

 /-* 



then fit) = — -T =U,+ U,k+ U^k" &c. ; 



for all values of k, whence JiUnf — xf as before. 

 33. Discussion of the Junction U^. 



h 

 fl-h 



By Art. 31. Uo = - — r when h is put eqifal to unity. Like the 



transient function Vo, discussed in Art. 27-, the quantity Uo is always 

 zero for values of t between and 1 ; but when t = 1 its value is 

 infinite, and thence its integral between the limits and 1 of / is finite, 

 viz. unity. 



