of Sturm's Theorem. 19 



it may easily be made out that between + co and L, each of 

 the quantities q v q 2 , q 3 , . . . q n will remain unaltered in sign ; 

 and between — oo and A also the same invariability of sign 

 obtains; and, moreover, between 4- go and L, and between A and 

 ""°° > {9i)> (fa)* > • • {9n-\)) (#J will be respectively greater than 

 1, 2 ... 2, 1. Consequently by virtue of the preceding theorem 

 between + go and L, and between A and — oo , D will always 

 retain the same sign as q v q 2 . q 3 . . . q n , and therefore no root of 

 fx will be contained within either such interval. And hence fx, 

 which is manifestly identical with D (the denominator of the con- 

 tinued fraction last above written), affected with a certain constant 

 factor, will retain an invariable sign within each such interval 

 respectively. Hence, then, the following rule. 

 Calling q lf q 2 , q 3 , . . . q n respectively 



a x x — b v a^x — 6 2 , a^x — b 3) ... a n x — b n , 

 if we form the 2n quantities 



V±l h±2 b 3 ±Z b n _,±2 b n ±l 



d\ &q #3 Q>n— 1 @"n 



the greatest of these will be a superior limit, and the least of 

 them an inferior limit to the roots of fx. 



The values of these fractions will depend upon the form of the 

 assumed subsidiary function <f>. Hence, then, arises a most 

 curious question for future discussion — to wit, to discover 

 whether in any case the subsidiary function can be so assumed 

 as that the superior limit can be brought to coincide with the 

 greatest, or the inferior limit with the least real root, supposing 

 that there are any real roots. I believe that it will be found that 

 this is always impossible to be done. Then, again, if all the roots 

 are imaginary, can inconsistent limits (evincing this imaginariness) 

 be obtained by giving different forms to the subsidiary function, 

 which would be the case if we could find that the superior limit 

 brought out by one form were less than the inferior limit brought 

 out by another, or the inferior limit brought out by one form 

 greater than the superior brought out by another ? If, as I 

 suspect, this also can never be done, then the general question 

 remains to determine for all cases the form to be given to the 

 subsidiary function, which will make the interval between either 

 limit and its nearest root, or between the two limits themselves, 

 a minimum. Thus, it appears to me, a fine field of research is 

 thrown open to those who are interested in the theory of maxima 

 minimorum, and minima maximorum, and one likely to lead to 

 unexpected and important discoveries. 



It may be asked how is the above rule to be applied if any of 

 the leading coefficients in <b(x), or of the successive residues of 



C2 



