Mr. J. J. Sylvester on the New Rule of Limits. 213 



a x x + b\ = 

 a 2 x + b 2 = 



/*3 + 



I 



^2 



6^0? + &! = 



a 9 x-\-b 2 = 

 a»x + bo = 



-A*i 



1 



" 2 + * 



fl i _ 1 a? + ^_ 1 = (-) l *-V t _ 1 + 



ax+b. 



/*i-l 





-r-^-i+ 





then (supposing fi } to have the same sign as a x ) the highest of 

 the values of x obtained from the first system, and the lowest of 

 the values of x found from the second system of these equations, 

 will be a superior and inferior limit respectively to the roots of 

 ^#=0; and so for all the rest of the equations 



AW-ojfrM>;. ..(/)*=o, 



excluding those of the first degree. 



It will be seen that the theorems contained in the observa- 

 tions (3) and (4) combined (which presuppose the statements 

 made in observations (1) and (2)), contain between them the 

 theorem given in the last Number of the Magazine, but rendered 

 in one or two particulars more simple and precise, and, as it were, 

 reduced to its lowest terms. In the whole course of my expe- 

 rience I never remember a theory which has undergone so many 

 successive transformations in my mind as this very simple one, 

 since the day when I first unexpectedly discovered the germ of 

 it in results obtained for quite a different purpose. In fact, it 

 never entered into my thoughts that in so beaten a track, and 

 in so hackneyed a subject as that of finding numerical limits to 

 the roots of an equation, there was left anything to be discovered; 

 and my sole merit, if any, in bringing the new rule to light, 

 consists in having been able to detect the presence and appre- 

 ciate the value of a truth which fortune or providence had put 

 into my hands. 



7 New Square, Lincoln's Inn, 

 August 6, 1853. 



