ON QUADRATURES AND INTERPOLATION. 329 



the differential tlieorem of expansion wLicli usually beai's liis name, and 

 which may be symbolically written as 



h '-- 



(/) (.1) + /() = £ ''•' . ^ {x). 



It is worth while to notice the two points : that he obtained it as a limit, 

 and that he obtained it indirectly, the assumptions of- continuity and con- 

 vergence being tacit.* Taylor does not proceed to discuss these points, 

 which, however, are now well known to all who have read Cauchy's 

 observations on the theorem. 



The full consequences of Taylor's two theorems appear to have been 

 first stated by Arbogastj in the symbolical fox'm — 



F(l -H A) . « = F£ '''^. «+ 

 When F is any function whatever, but is applied to the symbols of opera- 

 tion only, viz., A and li ——, and the resulting operation applied to ii. 



ctx 



Subject to suitable interpretation in the case of negative and fractional 

 values, this formula really includes the whole of the ordinary theoiy of 

 interpolation and quadratures. 



As regards fractional interpretation, since (1 + A)"" . «q := ?(,,., a 

 fractional index applied to (1 + A) simply means an interpolated value. 

 In the same Tvtiy a negative index applied to (1 + A) merely means a 

 preceding value. 



A fractional index applied to A has no useful meaning at all, being 

 indeterminate § ; a negative index also, strictly speaking, gives indeter- 

 minateness, which, however, is removable, within limits. For we have 

 to interpret A~' so that A . A~i ti,. =^ n,.. This is satisfied if A~i repre- 

 sents the indefinite series ending with m,,_2 + «,._,, and the use of 



this between limits gets rid of the indeterminateness. A more exact and 

 intelligible statement of this is that A"', standing by itself, is either 

 indeterminate or infinite ; but A~' u^ — A'^zj^ is perfectly determinate, 

 and stands for 



■w, + «,+i + v,-2 + ■'•i.-i- 



In this respect A~' is strictly comparable with /dx, and. in fact, we have 



* See the Methodus Incrementorum directa et inveisa, auctore Brook Taylor, 

 LL.D. . . . London, 1716. 



t See his Calcul dex BeHvations, Strasburg, 1800, pp. 313-352. See particularly 

 § 405, p. 3.51, Formula E. 



% It is worth while, following Arbogast, to distinguish this theorem from that 

 underwritten, in which F affects the whole expression that follows it : — 



F-|'(l+A)x(?)aT|= 6'"^'i: . V<px = (1 + A)"' X 'P<px. 



A little consideration will show that this is but another way of writing the identity 



Y <<p{x + mh)\ = V<p{x + mh) 



This is an important theorem, but a very different one, and it has no immediate 

 application to the object of this report. 



§ This is in fact a question of general interpolation. If the value of a function 

 is only known for integral values of n, there is no means of distingTiishing 



Vx from Far + <f> (sin «7'ir) .fx 

 which are whollj- different when r is fractional. This is but one example of a verj- 

 general truth. 



