6 MATHEMATICAL FORMULA AND ELLIPTIC FUNCTIONS 



If f(i/x) = o is put in a similar form, and the coefficients are all positive, 

 h is a lower limit of the positive roots. And with /(- i/x) = o, h is an upper 

 limit of the negative roots. 



1.255 Sturm's Theorem. Form the functions: 



f(x) = a Q x n + aix n ~ l + a 2 x n ~ 2 + .... + a n 

 MX) = /'(*) = naox n ~ l + (n- 

 MX) = -ft in f(x) = Q^(x) + 

 MX) = -R 2 in MX) = Q 2 (x) + 



The number of real roots of }(x) = o between x = xi and x = x z is equal to the 

 number of changes of sign in the series f(x), MX), / 2 (x), . . . when x r is sub- 

 stituted for x minus the number of changes of sign in the same series when x$ 

 is substituted for x. In forming the functions /i, / 2 , . . . . numerical factors 

 may be introduced or suppressed in order to remove fractional coefficients. 



Example : 



f(x) = x* 2x* 3# 2 + IQX 4 



MX) = 2 * 3 - 3* 2 ~ 3^ + 5 

 MX) = 9^ 2 -27^+11 

 MX) ".--to - 3 

 MX) = -1433 



+ 



+ 

 + 



3 changes 

 2 changes 

 i change 



Therefore there is one positive and one negative real root. 



If it can be seen that all the roots of any one of Sturm's functions are 

 imaginary it is unnecessary to calculate any more of them after that one. 



If there are any multiple roots of ]the equation f(x) = o the series of Sturm's 

 functions will terminate with f r , r < n. f r (x) is the highest common factor of 

 / and /i. In this case the number of real roots of f(x) = o lying between x = x\ 

 and x = x 2 , each multiple root counting only once, will be the difference be- 

 tween the number of changes of sign in the series /,/i,/ 2 , . . . .,/ r when xi and x 2 

 are successively substituted in them. 



1.256 Routh's rule for finding the number of roots whose real parts are 

 positive. (Rigid Dynamics, Part II, Art. 297.) 



Arrange the coefficients in two rows: 



X n 

 X n ~ l 



03 



05 



