140 Mr MURPHY'S SECOND MEMOIR ON THE 



and generally 



i;p.r..,=;r._,{i-f.^.*.'-ti^.<?!^<|±?l.f-&c.) 



= (-l)'"A"'.^^ '\ ^ ' ^^ '-.d){h). When h is put =0. 



' 1 . 2 m ^ ' ^ 



and by comparing the former integrals with the latter, the values of 

 ffo, «i, a-i, &c. are known, and being substituted give 



T._, = P,0(/^)-3P,A^.0(A) + 5P.A^^^±^^^±^.0(A) 



J. • <« • t7 



// being put =0, after the operations are performed. 



It should be observed here that the terms of this expansion are 

 perfectly independant of «, which only fixes the number of the terms; 

 hence this series may be continued to any number of terms, and we 

 shall always have ftT„.it^ = (p{x) provided x is any integer less than that 

 number, and consequently if the series be continued ad infinitum, the 

 equation will be true for all integer and positive values of x. 



Cor. Multiply both sides by if and integrate from t = to /f=l, 

 hence «^ (^) = ^ • <^ ('') + ^ . ^^^j;;^^^^^ A ^ . A 



+ ^-(x + l)(ar + 2)(x + 3)^ 1.2 '?>^ + *'C- 



when /* is put =0. 



This series may be used, not only for the integer and positive 

 values of x, but for any values which will not render it divergent. 

 (Vid. First Memoir, 'On the Inverse method of Definite Integrals,' 

 Art. 2.) 



