288 G. B. AIRY, ESQ. ON A PROOF OF THE THEOREM 



12. Or, two intersections may have been lost by the sliding of s and v beyond the 

 first and last limits of the diagram. In that case, if s, t, u, v, are all on the same side 

 of the line of abscissae, the disappearance of s, v, leaves t, u, subject to the reasoning 

 of Change [l]. If s, t, are above, and u, v, below, s may slide off to the left of the 

 diagram and v to the right, leaving t and u in the position proper for the final inter- 

 sections of P ( 00 ) and Q ( oo ), and then there will not necessarily be a root within the 

 limits of this diagram ; but on tracing the course of v into the next diagram (Stt to 47r), 

 which is exactly similar to this diagram (0 to Stt), it will be seen that v, which is now 

 below the line of abscissae, must rise above the line of abscissae, and consequently it 

 must cross the line of abscissae ; and therefore a root is ascertained. In like manner, in 

 tracing the course of s backwards into the preceding diagram ( — Stt to 0), s, which is 

 now above the line of abscissae, must form an intersection below the line of abscissae; 

 and therefore it crosses the line of abscissae, and therefore a root is ascertained. 



13. Change [s] may in all cases be resolved into combinations of Change [l] and 

 Change [2], and requires no special treatment. 



14. The general reasoning, applicable to all cases, is this. As r increases from 

 towards 00 , the intersections of the P-curve and the Q-curve must take place in pairs, 

 the two intersections which constitute any pair being, in the first instance, on the same 

 side of the line of abscissas. But adjacent intersections of P(oo) and Q ( 00 ) must in 

 all cases be on opposite sides of the line of abscissae. In the change from the former 

 to the latter state, one of the two intersections which constitute a pair must cross the 

 line of abscissae; and thus there must be as many roots as there are couples of adjacent 

 intersections of P( 00 ) and Q(x); that is, as many roots as there are different dia- 

 grams ; that is, there must be n roots. 



15. It appears to me that this demonstration of the existence of roots of an equation 

 is perfect and general. 



16. Perhaps some steps will be made in fully understanding the nature of the changes 

 of the intersections by consideration of an extreme case. Suppose that the equation is 

 of the lO'*" order, and suppose that the roots are all imaginary, the real parts being all 

 positive, and the proportion of the real part to the imaginary part being so nearly equal 



in all, that the values of Q for all will be included between — and — (imaginary term 



5 5 



Sir 9ir 



positive), and between and — (imaginary term negative). Then in the diagram 



o o 



(Stt to 47r) there will be, on giving a sufficient value to r, ten intersections, five of which, 

 on increasing the value of r, will cross the line of abscissa;, exhibiting five roots. In 

 like manner, in the diagram (l67r to ISvr), there will be ten intersections, of which five 

 will cross the line of abscissas, exhibiting five roots. In the other diagrams there will 



