234 THIRD REPORT— 1833. 



gree, under the ordinary rules of algebra, and which compels us 

 to consider different orders both ofinjinities and of zeros, though 

 when they are considei*ed without reference to their symbo- 

 lical connexion, they are necessarily denoted by the same sim- 

 ple symbols oo and : thus there is a necessary symbolical di- 

 stinction between (00)2, 00 and (oo ) , and between (0)2, and 

 (0) ; though when considered absolutely as denoting infinity 

 in one case and zero in the other, they are equally designated 

 by the simple symbols 00 and respectively. 



Though the fundamental properties of and co , considered 

 as the representatives of zero and infinity, are suggested by the 

 ordinary interpretation of those terms, yet their complete in- 

 terpretation, like that of other signs, must be founded upon the 



of the permanence of equivalent forms : thus, supposing, when x is a real 

 quantity, we can show that 



«^ = 1 + ^ + rr2 + r:V^ + ^^•' 



but that we cannot show in a similar or any other manner that 



1 . 2 1 . 2 . 3 ^^ °^*^"' 



then the equivalence in the latter case is assumed, by considering c* 

 as the abridged symbol for the series of terms 



1 + ^ V- 1 -Y72- TT2T3 + ^^- ' 



in other words, the form which is proved to be true for values of the symbols 

 which are general in form, though particular in value, is assumed to be true 

 in all other cases. 



It is true that such a generalization could not be considered as legitimate, 

 without much preparatory theory and without considerable modifications of 

 our views respecting nearly all the fundamental operations and signs of arith- 

 metical algebra ; but I refer with pleasure to this incidental testimony to the 

 truth and universality of this important law, from an author whose careful 

 and bold examination of the first principles of analytical calculation entitle 

 his opinion to the greatest consideration. 



Mr. Gompertz published, in 1817 and 1818, two tracts on the Principles and 

 Application of Imaginary Quantities, containing many ingenious and novel 

 views both upon the correctness of the conclusions obtained by means of ima- 

 ginary quantities and also upon their geometrical interpretation. The first 

 of these tracts is principally devoted to the establishment of the following 

 position: "That wherever the operation by imaginary expressions can be 

 used, the propriety may be explained from the capability of one arbitrary 

 quantity or more being introduced into the expressions which are imaginary 

 previously to the said arbitrary quantity or quantities being introduced, so 

 as to render them real, without altering the truth they are meant to express ; 

 and that, in consequence, the operation will proceed on real quantity, the 

 introduced arbitrary quantity or quantities necessary to render the first steps 

 of the reasoning arguments on real quantity, vanishing at the conclusion ; 



