248 THIRD REPORT — 1833. 



Signs of discontinuity are those signs which, in conformity 

 with the general laws of algebra, are equal to 1 between given 

 limits of one or more of the symbols involved, and are equal to 

 zero for all their other values. If merely conventiofial signs 

 were required, we might assume arbitrary symbols for this 

 purpose, attaching to them far greater clearness as diventical 

 marks, the limits of the symbol or symbols between which the 

 sign of discontinuity was supposed to be applied. Thus, we 

 might suppose ^T)a to denote 1, when x was taken between 

 and a, to denote zero for all other values ; ■^Da+^j, to denote 1, 

 when X was taken between a and a + b, and zero for all other 

 values ; and similarly in other cases. 



Thus, if 2/ = a X + /3 and y =^ of x + /3' were the equations 

 of two lines, and if we supposed that the generating point whose 

 coordinates are x and y was taken in the first line between the 

 limits and a, and in the second line between the limits a and b, 

 then we should have generally, 



y = -D; (a ^ + ^) + -D/ {«.' X + /30 (1.) 



the couvergency or divergency of the series which result. It is only, therefore, 

 when we come to specific values that a question will arise generally respecting 

 the character of the series : and it is only when we are compelled to deduce the 

 function which generates the series from the application of the theory of limits 

 to the aggregate of a finite number of its terms, that its convergency or diver- 

 gency becomes important as afi^ecting the practicability of the inquiry : in short, 

 it must be an erroneous view of the principles of algebra which makes the result 

 of any general operation dependent upon the fundamental laws of algebra to be 

 fallacious. The deficiency should in all such cases be charged upon our power 

 of interpretation of such results, and not upon the results themselves, or upon the 

 certainty and generality of the operations which produce them : in short, the 

 rejection of diverging series from analysis, or of such series as may become 

 divergent, is altogether inconsistent with the spirit and principles of symbolical 

 algebra, and would necessarily bring us back again to that tedious multipli- 

 cation of cases which characterized the infancy of the science. A verj' instruc- 

 tive example of the consequences of adopting such a system may be seen in the 

 researches of M. Liouville, which have been noticed in the note at p. 217. 



Lagrange in his Theorie des Fonctions Analytiques, and in his Calcul des 

 Fonctions, has given theorems for determining the limits between which the 

 remainder of Taylor's series, after a finite number of terms, is situated : and 

 the same subject has been very fully discussed in a memoir by Ampere, in the 

 sixth volume of the Journal de I'Ecole Poly technique. Such theorems are ex- 

 tremely important in the practical applications of this series, but they in no 

 respect affect either the existence or the derivation of the series itself. It is a 

 very common error to confound the order in which the conclusions of algebra 

 present themselves, and to connect difficulties in the interpretation and appli- 

 cation of results with the existence of the results themselves : and it is the in- 

 fluence of this prejudice which has induced some of the greatest modem ana- 

 lysts, not merely to deny the use, but to dispute the correctness of diverging 

 series. 



Messrs. Swinburne and Tylecote, the joint authors of a Treatise on the true 



