II. Suggestion of a Proof of the Theorem that every Algebraic Equation has a 

 Root. By G. B. Airy, Esq. Astronomer Royal. 



[Read Dec. 6, 1858.] 



In reading Professor De Morgan's demonstration of the existence of a root in every 

 algebraic equation, contained in a Paper lately communicated to this Society (a demonstra- 

 tion which is not to be mastered without very close attention), it occurred to me that a 

 different proof might be furnished, based on the same principle of comparing the values of P 

 and Q when r is very small and when r is very great, but dispensing entirely with the chain 

 of signs and its changes ; using, moreover, instead of the geometrical reference in Professor 

 De Morgan's proof, a geometrical reference of different character which admits of being 

 placed more distinctly before the eye ; and thus answering more perfectly to the etymological 

 meaning of " demonstration," namely, " clear exhibition." As the possession of this proof 

 has supplied to my own mind a satisfactory hold of a most important theorem which I have 

 sought in vain for many years, I have thought that it would not be presumptuous in me to 

 place it before the Society, in the hope that it may tend to satisfy, with other students of 

 Algebra, a want which I have myself felt so long. 



1. In order to take the subject in its utmost generality (which, however, scarcely alters 

 the steps of the demonstration), I shall, with Professor De Morgan, suppose the equation 

 to be 



ajg' + 6^r/-* + +m^^0, 



where the coefficients o^, h^, ... m^, may contain imaginary as well as real quantities. I 

 shall also follow his notation in thus changing the form of the terms : 



Let «[„=<* (''OS a + sin a v - O* 



6p = 6 (cos /3 + sin jS x/ - 1), 



TO^ = m (cos /* + sin /u v - 1), 



and also »", =» r (cos + sin 9 y^- i), 



where a, b, &c., a, j8, &c., n», fi, are known constants, and r and 6 are yet to be de- 

 termined. When the coefficients of the equation are entirely real, a, /3, &c., fi, are = 0. 



