285 



tenus partibus convocatis audias causam et appellatione remota 

 debito fine decidas faciens quod decreveris per censuram eccle- 

 siasticam firmiter observari. Testes autem qui fuerint nomi- 

 nati, si se gratia odio vel timore subtraxerint censura siinili 

 appellatione cessante compellas veritati testimonium perhibere. 

 Dat. Viterbii non. Octobr. Pontificatus nostri anno secundo." 



A vote of thanks was passed to the Rev. Mr. Thomson 

 and Mr. Brophy for their interesting donations, and to the 

 Rev. Mr. Abbott for his kindness in allowing the Papal Bull 

 to be exhibited to the Academy. 



The Rev. Dr. Graves read a paper on a generalization of 

 the Symbolic Statement of Taylor's Theorem. 



Lagrange was the first to show that Taylor's theorem ad- 

 mits of the symbolic statement, 



<p (x + h) = e hd * 00). 



The object of the present paper is to discuss the problem 

 of determining that more general symbol which shall have the 

 effect of changing x into any proposed function of x, such as 

 xfj (x), throughout the whole of a given function (x), which 

 is made the subject of the operation. 



The following synthetic course leads readily to the desired 

 end. As the effect of the symbol 



is to change x into a? + 1, it appears that 



d l d 



e W&) or ef' {x)dx ( a ) 



will change /O) into f(x) + 1; and consequently will have 

 the effect of changing x into i/i (x) where 



<K0 =/" a {/(*)+!) (4) 



