FOUNDATION OF ALGEBRA. 297 



dealing with them. These last can hardly receive due attention, unless 

 they are carefully distinguished from the previous and well-known cases 

 of the same kind ; which will be only done by adopting that system 

 of definitions which destroys the latter altogether. 



ADDITION. 



A theorem of M. Cauchy, which is well known to the readers of 

 Liouville's Journal, by the comparatively easy demonstration which 

 MM. Sturm and Liouville have there given of it, may be set in so 

 clear a point of view by the complete algebra, that I here add a de- 

 monstration of it. This theorem belongs essentially to the complete 

 system of algebra, as will be evident from its enunciation. 



As before, Z = {%, £) or ««W- 1 , means that Z is a line of the length 

 » inclined at an angle £ to the unit-line. 



Theorem. Let <pZ be any function of Z, and let Z = x + y J- 1 

 give <pZ = p +q J—\. Also, within the whole of 

 the figure ABCD, its contour included, let <pZ, (p'Z, 

 &c. never become infinite, x and y being the co-ordi- 

 nates of any point within it. Let any point be called 

 a radical point which makes cf>Z or <f>(x + y J — 1) = 0. 

 In carrying a point in the positive direction of revo- 

 lution round the contour ABCD, let the fraction " change sign by 

 /tossing through zero k times from + to - , and / times from — to + : 



but let it never change sign by passing through -, that is, let there 



be no radical point on the contour itself, and neglect altogether the 

 cases in which it changes sign by passing through oo . Then the number 

 of radical points contained within the contour is ^ (k — I). 



