190 Mr. Ilarveij on the Theory of Arbogast. [Sept. 



Of the particular departments of analysis to which the symbols 

 A, d, and J, respectively belong, and also of the peculiar opera- 

 tions of which they are destined to be the representatives, the 

 student may obtain ample information in the excellent transla- 

 tion of La Croix. And it shall now be my object to exhibit the 

 application of these symbols to the theory of Arbogast, and to 

 show how far it is capable to separate them from the symbols of 

 quantity. 



In the theory of finite differences, it is well known that if the 

 series 



/<_,. M_a M_, U M, Mj ..., U^ 



be assumed, that the excess of any one term over that which 

 immediately precedes it is called the difference of the latter, and 

 is denoted by the symbol A. 



On this hypothesis is formed the series of equations, 



u^ — i< = A u 



«.,— ",= A u^ 



7/3- ?/,= A w. 



"»— «„-.= ^ iln-. 



and which by evident transpositions become 



»,= u 4- A u 

 «,= w,+ A I/, 



Mj=: V.^+ A II., 



Or, separating the symbols of operation from those of quantity, 



«.= (1 -I- A) u 



«,= (1 + A) M. 



«3= (1 + A) u. 



«,= (1 + A)tt,_, 

 If now the value of u, in the first of these equations be substi- 

 tuted in the right hand member of the second ; and this new 

 value of the second substituted in the third, and so on ; there 

 will arise 



«,= (1 + A) u 

 «,= (1 + A)^ u 

 11,^ (1 + A)^ u . (1.) 



u = (1 + A)" ic 

 in each of which equations, the expression 1 + A is to be 

 regarded as having no other meaning than an abbreviated 

 expression for its developement, but which when expanded, and 



