548 



Mr de morgan, on various points 



APPENDIX. 



ON THE CONVERGENCY OF MACLAURIN'S series. 



A remarkable theorem was given by M. Cauchy in 1831 or 1832, and repeated in 1840, 

 (Exercices d' Analyse... Paris, 1840, 4to. pp. 269-279; Moigno, Lecons... Paris, 1840, Vol. I. 

 pp. 150— 16"1), which appears to have attracted* considerable attention, and to have met with some 

 objection. It is as follows : — The development of <px in powers of x continues convergent 

 from x m until x becomes equal to the least of all the moduli in the roots of the equations 

 <p<v co , (p'x = co : the modulus of a quantity, real or imaginary, being a positive quantity 

 r, when the quantity is exhibited as r (cos 9 + sin 0. /y/- 1). I think M. Cauchy means to 

 imply that divergence must begin when this limit is past : but I cannot find any express 

 statement to this effect. 



That this theorem is not generally true, may be at once established by the simple 

 instance <p,v = (1 + w) , which satisfies the definition of continuity cited in the preceding note, 

 and gives no finite real or imaginary root to either (f>x = co or (p'x = co . It ought then 

 to have a convergent development for all values of w ; but it is very well known to have 

 a divergent development for all values of x greater than unity. 



I shall show that this theorem cannot be true unless cp^x = co be introduced into the con- 

 ditions, for all values of n : and that both the direct and converse theorem may then be 

 established. I think it probable that the assent which M. Cauchy's enunciation has received, 

 arose from the impression that by a continuous function, between given limits, must have been 

 intended a function which remains finite and determinate between those limits, and all its dif- 

 ferential coefficients also. This seems to me to be the best definition of the term ; and if my 

 supposition as to the mode in which the imperfect theorem gained assent be correct, there is 

 good reason to think that, by express statement, this definition-)- might soon be established. 

 In this case it might be desirable to say that there is continuity of the n tb order, when between 



" In the Exercices M. Cauchy says " Parmi les the'oremes 

 nouveaux, que contiennent mes me'moires de 1831 et 1832, sur 

 la mecanique celeste, l'un des plus singuliers, et en meme 

 temps l'un de ceux auxquels les ge'ometres paraissent attacher 

 le plus de prix, est celui qui donne imme'diatement les regies 

 de la convergence des series fournies par le d6veloppement des 

 fonctions explicates, et rdduit simplement la loi de convergence 

 a la loi de continuity, la definition des fonctions continues 

 n'etant pas celle qui a etd longtems admise par les auteurs des 

 traitcs d'algebre, mais bien celle que j'ai adoptee dans mon 

 Analyse Algibrique, et suivant laquelle une fonction e3t con- 

 tinue entre des limites donne"es de la variable, lorsque entre ces 

 limites elle conserve constamment une valeur finie et ddter- 

 mine'e, et qu'a un accroissement infiniment petit de la variable 

 correspond un accroissement infiniment petit de la fonction 



elle-meme." I may also add that M. Coumot, in his excellent 

 work on the infinitesimal calculus (Paris, 1841) states the 

 theorem, in the form in which M. Cauchy gave it, and without 

 any doubt of its perfect generality. The definition of con- 

 tinuity here given supposes all values, real and imaginary, of 

 the variable. 



+ Any definition would be better than none at all. At 

 present it is (to me at least) a most difficult task, in reading a 

 work, first, to find out the author's definition of continuity, 

 next, to find out whether he sticks to it. M. Moigno, in his 

 enumeration of cases under M. Cauchy's theorem, sets down 

 (l + x) m "~, generally, as discontinuous when x = \ : here is an 

 implied definition of continuity differing from that on which 

 the theorem is constructed. 



