1820.] Mr. Hat-vet/ on the Theory of Arbogast. 191 



the value of u restored, will furnish the absolute value of the 

 function ?/„, with all its orders of differences. For example, 

 La Croix, at p. 4, of his " Traite des Differences," observes that 



n (n - 1) . „ „ f„ _ 1) („ _ 2) ^ , 



«„= » + » A « + \ ^ A- u + ■ 1 a.3 — A' «' 



4- A" // ; and it is evident, that if the conditional equation 



w.= (1 + A)- a 



be developed, and the function u restored to the symbols of 

 operation, the same result will be obtained. 



Arbogast arrives at a similar conclusion, pp. 375, 377, " Du 

 Calcul des Derivations," nearly as follows : 



Let E° ?/, E' M, E- ?< E» M 



represent consecutive states of the function u, so that 



E° u = u 

 E' u = ?/, 

 E* u = M, 



E" « = u„ 

 then since E' u, or E » = «', and that by the theory of differ- 

 ences u^ = << + A u, it follows that E ^( = ?/ + A ?< ; and by 

 separatinsc the function u from the symbols of operation. 



] ^ E = 1 + A,* ^ 



Whence 



E-= (1 + A)" ; 

 and restoring the function u to the symbols of operation 

 E" X « = (1 + A)" X u 



Or, 



«„ = (1 + A)"?/, 

 the same as above deduced. 



By an equally ingenious process, the same able mathematician 

 arrives at the value of A" u, as follows : 

 Since E = 1 + A, it follows that 



A = E - 1, and A" = (E - 1)". 

 Hence, by restoring the function u to the symbols of opera- 

 tion, we have 



A" « = (E - 1)"?/; 



the latter member of which developed by the binomial theorem, 

 and}^ restored to the symbols of operation, gives 



A"« = E-»- «E-' « + "';'~'' e-"-» ± £"«; 



and since by the hypothesis E" m = ii^, E"~' u = i<„_, 



E""" «' = M„_.2 E° ?* = u, there arises by substitution, 



A. n (n — 1 ) 

 M = w„ - n ?^._, + ^p-g-' »„_, ±11. 



* The symbol = docs not denote an «Jso/u/e eyjiaWj/ between E, A, and 1, but 

 IS siin|tlY used to connect two modes of cxine-sion arbitrarily made equivalent, — 

 (See Woodhouse'8 Principles of Analytical Calculation, p. Vi3.) 



