w 



[ 121 3 



XVI. On the Evaluation of Vanishing Fractions, with some Sup- 

 plementary remarks on Newton's Rule. By J. R. Young, for- 

 merly Professor of Mathematics in Belfast College*. 



HENEVER a fraction ~-^ takes the form - for some par- 

 f(x) * 



F (x) F (x) 

 ticular value a of x } we may replace it by > 1V ; y y , &c, in 



succession, till we arrive at a fraction in which numerator and 

 denominator do not both vanish for x = a. And it is plain that 

 these fractions will remain unaltered though for F 2 (%), F 3 (#),F 4 (a?), 

 &c, and at the same time fovf 2 (x),f s (x),f 4 (x), &c. we substitute 



1^*) F3W _F 4 W_ 

 2 ' 2.3' 2.3.4' '' 

 and 



2 ' 2.3' 2.3.4' 



&c. 



This being so, I think that whenever F(a?) and/(#) are rational 

 polynomials, it will be more systematic and at the same time more 

 easy, to compute the value of the vanishing fraction as in the 

 following examples^ a step of the work on the left and a step on 

 the right being taken alternately. It will be readily foreseen 

 that, although no expressions of higher degree than the third are 

 here taken, these being fully sufficient for the purpose of illus- 

 tration, yet the higher the degree, and the larger the number a } 

 the greater is the facility of this method of calculation as com- 

 pared with that usually employed. 



1. ~F{x) = x 3 -2x 2 -x + 2, f(x)=z s -7x + 6: to find S?). 



J\6) 



1_2 -i +2(2 

 2 0-2 

 0-1 





1 ■ 

 2 

 2" 



-7 +6(2 



4 -6 

 -3 



2 4 

 2 3 





2 



4 



8 

 5 



1 



\m. 



3 





* " tm ' 



5 





* Communicated by the Author. 



