236 THIRD REPORT — \833. 



If we assume a to denote a finite quantity, then 

 (1.) a + = a, and a + oo = + oo . 



Consequently does not affect a quantity with which it is 



connected by the sign + or — , whilst co , similarly connected 



with such a quantity, altogether absorbs it. 



(2.) axO = 0, axoo =oo;--- = co and — = 0. 



00 



It is this reciprocal relation between siero and infinity which 

 is the foundation of the great analogy which exists between 

 their analytical properties. 



(3.) If these symbols be considered absolutely by themselves, 

 without any reference to their symbolical origin, then we must 



consider ~ = \ and = 1. 



CO 



But if those symbols be considered as the representatives 

 equally of all orders of zeros and infinities respectively, then 



~ and may represent either I or « or or co , its final 



form and value being determined, when capable of determina- 

 tion, by an examination of the particular circumstances under 

 which those symbols originated. The whole theory of vanish- 

 ing fractions will depend upon such considerations. 



Having ascertained the principal symbolical conditions which 

 and CO are required to satisfy, we shall be prepared to con- 

 sider likewise the principle of their interpretation. The exami- 

 nation of a few cases of their occurrence may serve to throw 

 some light upon this inquiry. 



Let us consider, in the first place, the interpretation of the 



critical values 0, oo and ~ in the formulae which express the 



values of x and y in the simultaneous equations, 



a X + b y = c \ 

 a' X + b' y ■= c' J 

 In this case we find 



^ t:- and y = 



X 



'•4^-!} -'{I'-l.}' 



„' „ „' 



If -7- = -jT, — = — 7) and therefore -z- — -n, then a: = -^ 

 b a a bo 



and y = -Q- 



