THAT EVERY ALGEBRAIC EQUATION HAS A ROOT. 285 



maximum of these diminished ordinates has a value nearly independent of r; and then we 

 can contemplate with ease the relations of the two curves even when r is made indefinitely 

 great. 



5. As it is our object to prove that P and Q may become = at the same time, it would 

 at first seem best to discover the law which determines the intersections of the P-curve (or 

 the curve connecting all the summits of the ordinates representing the value of P) with the 

 line of abscissae, and in like manner to discover the law which determines the intersections of 

 the Q-curve (or the curve connecting all the summits of the ordinates representing the 

 values of Q) with the line of abscissae ; and then to shew that intersections of the two curves 

 with the line of abscissae must at some time fall on the same point. But I believe that it will 

 be easier to consider the intersections of the P-curve with the Q-curve, and to shew that, 

 with some value of r, one of these intersections must have moved across the line of abscissae. 

 The process will therefore be. 



To exhibit the forms of the P-curve and the Q-curve when r = 0. 



To exhibit the forms of the P-curve and the Q-curve when r is indefinitely great. 



To infer from these the general character of the formation of intersections, and change 

 of the points of intersection, of the two curves. 



To shew that, in some part of this change, one of the points of intersection must 

 have crossed the line of abscissae. 



6. When r = 0, the P-curve is a straight line whose ordinate = m cos fx. When r is 

 indefinitely great, the first term only in the value of P is sensible (as being indefinitely greater 

 than the others) ; and the P-curve, with its ordinates diminished as is mentioned in Article 4, 



is a Ime of cosines, or a line of sines drawn back through — . And when r = 0, the Q-curve 



is a straight line whose ordinate = m sin /u : when r is indefinitely great, the first term only in 

 the value of Q is sensible, and the Q-curve, with its ordinates diminished, is a line of sines. 

 In conformity with these indications, the following diagrams are drawn : where P (0) and 

 P (oo ) denote the P-curves corresponding respectively to r = 0, r = oo ; and Q (0) and Q (os ) 

 denote the Q-curves for r = 0, r = co . I have supposed that m cos n and m sin m are both 

 positive, and that m sin ix is smaller than m cos n ; but it will be seen in the demonstration 

 that the relation of these magnitudes is unimportant ; the two values may even coincide ; the 

 only condition which cannot be admitted is, that P(o) and Q(0) intersect at a single point, 

 which indeed can never happen. 



