FOUNDATION OF ALGEBRA. 299 



By our hypothesis m is an integer, and there are m radical points in 

 one, answering to m equal roots of <pZ=0. Also, B being (b , fi ), 

 we have 



B B m = b^r™ {cos (mp + /3 ) + sin (m P + /3 ) . J- 1 } . 



whence * = cot (mp + /3 ). 



Now while p goes through a whole revolution, mp + fi passes from 

 (i to 2mir + /3 through »« complete revolutions, and changes sign 2m 

 times from + to - passing through each time : but it never changes 

 from — to + except by passing through oo . Hence k = 2m, 1=0, 

 l (k - I) = m, which verifies the theorem, since there are m roots within 

 the contour. 



Next, let the whole figure ABCD be divided into an infinite number 

 of infinitely small figures, with no other limitation than that no radical 

 point is to fall upon one of the lines of division: and let a point move 

 round each of the infinitely small figures in the positive direction of 

 revolution. It is clear that the expression ^ (2A - 2/) will not be altered 

 if we remove all the internal division lines and leave only the external 

 contour ABCD : for each internal line is described by two points moving 

 in opposite directions, and wherever one point adds a unit to 2&, the 

 other adds one to 2£ Hence the value of Ik - 2/ can be found by 

 finding that of k — I for the external contour only. 



If we suppose cpZ to be rational and integral, say = A Z n + AiZ"- J + ..., 

 and if we make the contour in question a circle with the origin as a center, 

 and a radius so great that the highest term A Z" need be the only one 



retained, we find that - = cot (»£ + a ), which gives, as before, in a 



revolution, k = 2n, I - 0; whence the whole number of roots of (pZ=0 

 is neither more nor less than n. 



It is easily deduced from the preceding that the number of real 

 roots of <px = lying between x = a (the less) and x = b the greater, is 

 the number of vanishing changes of sign from — to + which take place 

 while x passes from a to b in the quotient of 



>*- 0".r£ 4 c^x-V--- ... divided by <j>'x. - <p'"x^- + $>x 



2 T 2.3.4 J -r • T 2.3 r 2.3.4.5 



