THAT EVERY ALGEBRAIC EQUATION HAS A ROOT. 287 



Change [2I. Or the P-curve has intruded upon the Q-curve (or vice versa) in two 

 sinus so as to produce four intersections ; of which two have been 

 subsequently obliterated. 



Change [3]. Or a greater number n of such sinus have been formed, producing 2n 

 intersections; and 2n— 2 intersections have been subsequently obliterated. 



I shall now consider the movement of the intersections while these formations and 

 changes of sinus were going on. 



9. In Change [l]; when one curve begins to intrude upon the other, it first intrudes 

 on it by simple contact. If that contact occurs exactly on the line of abscissae, the in- 

 vestigation is terminated; P and Q vanish together, and a root has been found for the 

 equation. But if the contact does not occur on the line of abscissae, it occurs (say) 

 aibove it ; as the intrusion advances, the simple contact is changed into two intersections, 

 both above the line of abscissae. But one intersection of P( co ) and Q ( 03 ) is below 

 the line of abscissae. How can this have been formed ? It can only have been formed 

 by the downward-travelling of that intersection of the P-curve and Q-curve which was in 

 fact the lower end of the sinus of intrusion, till it crossed the line of abscissae to its 

 lower side. At one instant therefore this intersection was on the line of abscissae. At 

 that instant, P and Q vanish simultaneously, and a root is found for the equation. 



If the first contact of the two curves had occurred on the lower side of the line of 

 abscissae, the upper of the two intersections must have crossed the line of abscissae to 

 the upper side, in order to form the upper intersection of P(oo) and Q(oo); and the 

 conclusion is the same. 



It will be remarked that it is not necessary that the simple contact and the first 

 formation of the sinus should commence in the diagram which is before us. They may 

 have commenced in the diagrams to the right or to the left of that which is before us, 

 and the intersections may then have travelled sideways into this diagram. 



10. Change [2] may be effected in three ways. Either, when the two sinus have 

 been formed, one of them may afterwards have been destroyed by the withdrawal of the 

 protrusion that formed it; which leaves every thing in the same state as if it had never 

 been formed, and therefore leaves the other sinus in the state of Change [l]. 



11. Or, when two sinus have formed four intersections, s, t, u, v; two of these may 

 have been lost, by the union of t, u, into a point of simple contact, and the subsequent 

 separation of the curves there. In that case, if s, t, u, v, are all on the same side of 

 the line of abscissae, the union of t, u, and subsequent separation of curves, leave s, v, 

 on the same side, and the reasoning of Change [l] applies. If s, f, are on one side 

 and u, V, on the other side of the line of absciss®, the approach o{ t to u forms an 

 intersection upon the line of abscissae, and a root is found. 



Vol. X. Part II. ' 37 . 



