328 REPORT— 1880. 



Witli quadrature, it is rather different. The ordinary formulae of 

 quadratures presuppose the relation of ordinate and abscissa, and any 

 variation of this supposition, such as a transformation to polar co-ordi- 

 nates, requires a corresponding change in the formula. 



In connecting integration, regarded as the inverse of differentiation, 

 with integi-ation viewed as the limit of summation, the fundamental 

 assumption is continuity, and that in a very restricted and special sense. 

 Its applications to interpolations • aud quadratures also presuppose con- 

 vergence, and unless both continuity and convergence are secured, any 

 arithmetical conclusions drawn from our processes must fail for want 

 of evidence of their application, when strictly considered. For rough 

 practical purposes, a great inany processes are useful in ordinary cases, 

 which will not bear carrying to extremes. 



IV. — Theorems of Finite Diffeeences. 



Section 1. — Co»i))to7iformitles of direct interjMlation and quadrature hj ordinary 



differences. 



If Uq, «!, 1(2 11,1 be any n + \ consecutive series of numbers 



or homologous quantities whatever, and if we difference them so that 

 w,.^i — M^ = Aw,., it is well known, and can be easily proved by induction, 

 that the operative symbol A is subject to the ordinary laws of algebraical 

 combination, and that, provided the subject of operation does not alter, 

 it may be treated as an algebraical constant. Moi-eover, since a con- 

 sequence of this is 



«;•+! = (1 + ^) • «r 



the suflSxes also follow the index law, and 



-«,.+, = (1 -I- A )".«,. 



The only limitations are that q and r shall be positive and integral, and 

 that r + q shall not exceed n. With these limitations, the expansion is 

 identically verified. So far, no assumption is needed concerning equi- 

 distant variations. 



The theory of finite differences assumes that ii^, -z*, m„ are 



successive states of a function of a variable increasing by equal incre- 

 ments, so that if Uq =: (pz, and if h be the increment of .r, u,. z= (p (^z + rJi). 

 Then by mere induction we obtain 



<^ (z -f nh) =,pz + n A.<j,z + ^ ^^ ~ ^^ A^ . fz 



, M (w — 1) (w ~ 2) . 3 ^ , ^ 



+ ^ -^ ,^3 ' A^.i\>z + * 



This, with a slight difference of notation only, is Brook Taylor's theorem 

 in finite differences, which would now be expressed symbolically by 



(l> {z + nil) = (1 + A)" . <l>z 



Then, making the further assumptions of continuity and convergence, in 

 exactly the same way as the expansion of c* is deduced as the limit of 



the binomial expansion (1 + -^ j when r is made infinite, Taylor deduced 



* It is worth while to compare this with the corresponding geometrical expression 

 given by Newton iPrincijna, Book III. Prop. xl. Lemma 5, Case 1, 3rd ed. 1726, p. 

 486). , 



