REPORT ON CERTAIN BRANCHES OF ANALYSIS. 239 



The second member of the equation 



— ~~ + T2 + «3 + • • • • 



a — b a a'- a" 



preserves the same form, whatever be the relation of the values 

 of a and b, and the operation, which produces it, is equally prac- 

 ticable in all cases. As long as a is greater than b, a~b is po- 

 sitive, and there exists, or may be conceived to exist, a perfect 

 arithmetical equality between the two members of the equa- 

 tion. If, however, a = b, we have -jr upon one side and the 



sum of an infinite series of units multiplied into — upon the 



other, and both the members are correctly represented by oo ; 

 but if a be less than b, we have a negative and a finite value 

 upon one side of the equation, and an infinite series of perpe- 

 tually increasing terms upon the other, forming one of those 

 quantities to which the older algebraists would have applied 

 the term plus quam infinitum, and which we shall represent by 

 the sign or symbol co . It remains to interpret the occurrence 

 of such a sign under such circumstances. 



The first member of this equation — — -r is said to pass 



through infinity when its sign changes from + to — , or con- 

 versely : its equivalent algebraical form presents itself in a se- 

 ries which is incapable of indicating the peculiar change in the 



nature of the quantity designated by — — -y , which accompa- 

 nies its change of sign. The infinite values, therefore, of the 

 equivalent series (for in its general algebraical form, where no 

 regard is paid to the specific values of the symbols, it is still 

 an equivalent form,) is the indication of the impossibility of ex- 

 hibiting the value of ^ in a series of such a form under 



such circumstances. 



Let us, in the second place, consider the more general series 



for {a — by, or 



„ f, b ^ n{n- 1) b^ 



n {n -I) in - 2) 1% & 1 



The inverse ratio of the successive coefficients of this sei'ies 



