of Sturm's Theorem. 17 



If integer numbers be used throughout (so that accordingly the 

 (w) series is that made use of), the total number of multiplica- 

 tions will in general be n + 2(n— l)*or 3/i— 2; the old method, 



as previously stated, would require n. — ~— multiplications; for if 

 we call any one of the Sturmian functions 



A a?i + A,.a?i-i + A 2 .a?i-2+ ... + A 2 , 

 we shall, using the most abbreviated method of computation, 

 have to calculate successively 



tf.Ao + Ajj tf^Ao+Aj) +A3, &c, 

 giving rise to 1 operations (but it must be admitted with the 

 practical advantage of the use of a constant multiplier) ; and as 

 \C) may take all values from n to 1, the total number of such 



n-\- 1 

 operations will be n . . When 71= 4, 



„njl =8re _ 3 . 



Consequently (if it be thought necessary to adhere to integers 

 throughout), for values of (n) not exceeding 4, the old method 

 would be probably the more expeditious. 







Addendum. 



On a method of finding Superior and Inferior Limits to the real 

 Roots of any Algebraical Equation. 



The theory above considered has incidentally led me to the 

 discovery of a new and very remarkable method for finding 

 superior and inferior limits to the real roots of any algebraical 

 equation. Suppose in general that 



then it is easily seen that 



D = M 1 .M 2 .M 3 ...M„, 



where 



M i = ?i> M 2 =g 4 + -, M 3 =^ 3 + 1F . . . M. n =q n + 



qi > ^^ M.'-'^" *"' Mni 

 In general let any numerical quantity within brackets be used 

 to denote its positive numerical value; so that, for instance, 

 whether q~ + 3, (q) will equally denote +3. 



* If all the extraneous factors are units, the number of multiplications 

 (like that of the additions) would be 2n — 1, and not 2n, as inadvertently 

 stated in the preceding number of the Magazine. 



Phil. Mag. S. 4. Vol. 6. No. 36. July 1853. C 



