55 



Euler* first gave the integral of this equation, and has since 

 been followed by many other authors. 



1 + A X + B x^ ■}- &c. - - - + P X" is always re" 

 solvable, as is well known, into simple or quadratic factors^ 



The quadratic factors not resolvable into simple factors are, as 

 is also well known of the form 



(1 — «a:)H- /3'^' (2)- 



when 0=0, there are two equal simple divisors (1 — ax^. 

 now according to Euler, and I believe all authors who have 

 since written on the subject (to the last of whom, Lacroix, I may 

 particularly refer) the part of the integral corresponding to the 

 quadratic factor (2) is 



c, e^'^cos l3x -{- c^e'^ sin /Svr (3) 



where c, & c, are constant arbitrary quantities, and the part of 

 the integral corresponding to the two equal roots is 



C gax _|_ c\ c^^ X (4) 



now it would at first view naturally be expected that the ex- 

 pression (4) would be deduced from the expression (3) by tak- 

 ing the limits when 0=-o. 



But in this case we obtain only c , e^ instead of e"^ (c , + c , a:) 

 Here the application of limits seems to fail entirely. 



The expression (3) is true whatever definite value we assign 

 to 0, and yet is not true of the limit to which this quantity 

 approaches, when /3 is indefinitely diminished and becomes evan- 

 escent. 



This, which certainly appears a sort of paradox, may be thus 

 explained : 



* Nov. Comm. Acad. Pet. Tom, 3. 



