350 MEMOIRS OF THE AMERICAN ACADEMY. 



I define the first difference of a function by the usual formula, 



(113.) A<px = (p(x-{- Ax) — (px, 



where A x is an indefinite relative which never has a correlate in common with 

 So that 



(114.) x,{Ax) = z-+Ax = x-^Az. 



Higher differences may be defined by the formulae 



(115.) 



An-Z =0 if tl > I . 



A 2 - (px = AAz = q>(x + 2.. Ax) — i.(p(x + A a?) + (px , 



A 3 fx = AA 2 -z = <p(x -f- 3-A#) — 3-<p(# + 2-Aa?) -|- 3-<p{ x + A a?) — 9 X 

 (116.) An-ya? = <?(# + n.Aa?) — xt.<p(x + (it — i).Aa?) 



+ n ' (n 2 ~ l} -y(g + (K — a). As) — etc. 



The exponents here affixed to A denote the number of times this operation is to 

 be repeated, and thus have quite a different signification from that of the numerical 

 coefficients in the binomial theorem. I have indicated the difference by putting a 

 period after exponents significative of operational repetition. Thus, m 2 may denote a 

 mother of a certain pair, m 2 - a maternal grandmother. 



Another circumstance to be observed is, that in taking the second difference of x, if 

 we distinguish the two increments which x successively receives as A'a; and A"x, then 



by (114) 



(A'x),{A"x) = 



If A a? is relative to so small a number of individuals that if the number were 

 diminished by one A n -(px would vanish, then I term these two corresponding differ- 

 ences differentials, and write them with d instead of A. 



The difference of the invertible sum of two functions is Jhe sum of their differ- 

 ences; for by (113) and (18), 



(117.) A(<jpa; -f- yx) = (p(z -f- Az) -\- y(z + Az) — <pz — yz 



: tp(x -\- Ax) — (px -|- y(z + Aa;) — y/z = A(pz + Aya; . 



