300 PROFESSOR DE MORGAN, ON THE FOUNDATION OF ALGEBRA. 



y being an infinitely small positive constant : provided that neither a nor b 

 is a root. This residt, however, would be of little general use: in fact, 

 this theorem of Cauchy requires an examination of the contour which 

 is equivalent in that which must be made of the axis to find the real 

 roots. It makes the examination of a line equivalent to that of the 

 whole included space ; but does not profess to help in that examination. 

 But in the important case in which the contour is a rectangle with sides 

 parallel to the axes, or when it is desired to find all the roots of the form 

 x + y y/ — 1 in which x lies between given limits, and y between other 

 given limits, this theorem is a complete reduction of the question to 

 that of finding the number of real roots of four equations which lie 

 between given limits, one pair for each equation. It thus supplies the 

 theoretical desideratum which Fourier and Sturm have left. 



A. DE MORGAN. 



University College, London, 

 October 12, 1841. 



