96 Mr. J. R. Christie on the Extension of Budan's Criterion 



us, a generation which passes over our grave is sufficient 

 to cause these titles not to be remembered; the facts are 

 quoted, the authors are forgotten. 



The works of Richter, as we have seen, belong to these two 

 distinct classes, and if it is true that a few words should suffice 

 to sum up the entire life of a celebrated man, that of Richter 

 is altogether summed up in these words (taken from the Wis- 

 dom of Solomon, xi. 22) which he placed as an epigraph at the 

 head of all of his works which treat of chemical proportions: 

 " God made all things, in measure, and number, and weight." 



XVI. On the Extension* of 'Budan's Criterionfor the Imaginary 

 RootS) and a new Method of effecting the Separation of the 

 nearly equal Roots of a numerical Equation. By James 

 R. Christie, Esq.\ 



T5UDAN has shown that his criterion of the presence of 

 imaginary roots only fails when, in the pair of roots 



a + /3 V* — 1, a is a positive proper fraction and /3 is less 

 than *5, on account of the effect of his reciprocal transforma- 

 tion being that of converting these roots to the new form 



- ~g + 02 or a i ± & V -1» 



wherein a y must, in the failing case, be less than unity. 

 In the reduced reciprocal equation these roots become 



and they may, as before, be shown to be imaginary unless /3j 

 be less than *5. 



If we suppose a to be not greater than /3, then — — will be 



the least value of the fraction /3 X ; but /3 is less than *5, conse- 

 quently this value of /3 X must exceed unity. It appears there- 

 fore that, in the case of a not greater than /3, the condition upon 

 which the failure of the criterion depends, ceases to exist in 

 the roots as they appear in the first reduced reciprocal equa- 

 tion. The same will hold true if a does not exceed /3 y 3, 

 since the least value this condition allows for /3j is '5. 



Let us now see in what manner a. and /3 enter into the se- 

 cond reciprocal equation. 



* It is proper to mention that, in 1840, I pointed out the practical ap- 

 plication of this method, in an example which was casually brought under 

 my notice, to my friend and colleague Mr. Davies, who considered the 

 then crude remark as of sufficient importance to be inserted, with the 

 example, in his last edition of Hutton's " Course of Mathematics." — 

 J. R. C. 



\ Communicated by the Author. 



