THAT EVERY ALGEBRAIC EQUATION HAS A ROOT. 289 



be no intersections at all. As the value of r is indefinitely increased, the intersections 

 will spread away to the right and left, till finally there are two intersections and no 

 more in every single diagram. 



17. The existence of real roots of the equation depends on the contingency of in- 

 tersection taking place upon the line of abscissae at the point where = 0; or, supposing 

 all the coefficients of the equation real, and therefore a = 0, it depends on the intersection 

 taking place at the left-hand extremity of the diagram. Now with real coefficients, Q 

 always vanishes when = 0; or, the Q-curve always intersects the line of abscissae at the 

 left-hand extremity of the diagram. If the last term of the equation be negative, P(0) 

 will be below the line of abscissae; and the P-curve, in the course of its change to the 

 state of P(eo), must pass through the left-hand extremity of the diagram, and therefore 

 must make an intersection there with the Q-curve; and thus there will be a real positive 

 root; as is otherwise abundantly known. The same reasoning applies in all respects when 

 = 7r (the order of the equation being always supposed even): but in the interpretation 

 of the result there will be this difference, that the root of the equation instead of being 

 rcosO, as in the case just considered, will be rcosTr, or —r: and thus the equation will 

 have a real negative root as well as a real positive root : as is also well known. 



G. B, AIRY. 



Royal Observatory, Greenwich, 

 November 9, 1858. 



37—2 



