186 Mr DE morgan, ON INTRODUCING DISCONTINUOUS CONSTANTS 



forced attention to the subject. And even then, the discontinuous 

 constant was only a new fundamental symbol, inserted in its proper 

 place, in such form and manner as what I may call discontinuous 

 investigations shewed to be necessary. In the method which I propose 

 to explain, discontinuity not only appears in its proper place, but 

 with its proper symbol. 



When n terms of the series can be expressed in terms of 71, the 

 supposition n = ac will generally point out, in one way or another, 

 Avhether any, and what, discontinuity exists. The method which I 

 proceed to explain, while it depends for its strictness upon the passage 

 from a finite to an infinite number of terms, does not require the 

 actual expression of m terms as a condition of practicability. 



As usual, let (f>x, tp^x, &c. represent the results of successive func- 

 tional operations ; the symbol <px admits of two distinct characters, in 

 the periodic and non-periodic cases. Either (p'x = x, for a finite whole 

 value of n, or for no whole value whatsoever, except in the extension 

 n = 0. In cases where <px is not periodic, it has this peculiarity ; 

 that (f>"x, whatever may be the value of x, will either increase (with 

 x) without limit, or will, for successive whole values of w, give a 

 series of approximations to »« different limits which are severally roots 

 of (p^x = X. I am speaking of positive or negative functions, and of 

 real roots. With this proposition my only concern here is as to the 

 case where (p'x has one limit, in which case it evidently must give 

 (pL =^ L, L being the limit in question. And this proposition is 

 already well known in every part of mathematics. For instance, most 

 direct methods of successive approximation depend upon the use of 

 Taylor's Theorem, in a manner which will be recognized in the fol- 

 lowing particular case. If 



(px , . d(px 



^^• = ^■-0^ "^^'^'^ 1'-'=-d^' 



then the limit of successive operations gives a root of \px = x ; that is, 



either of (px = 0, or of -;— = 0. 



^ <p X 



