VI. Supidement to a Proof of the Theorem that every Algebraic Equation has a 

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



[Read Bee. 12, 1859-] 



18. In lately offering to the Society a Proof of the Theorem that every Algebraic 

 Equation has a Root, I assumed (as a thing to be proved by the process) that a root might 

 be expressed in the usual way, by the aggregate of two terms, of which one is real, and the 

 other is imaginary, involving the symbol v^ — l as an ostensible multiplier. And the proof 

 went by these principal steps. Changing the notation there used for one which is more 

 familiar to us, supposing all the coefficients real, and supposing (for ease of writing) that 

 the equation is of the 8th order, instead of the nth ; and assuming the equation to be 



ai^+p.w'' + q.aP-\-r.ar' + s.x^ + t.af*.\-v.w^ + w.!C + z = 0', 

 then, adopting the expression p (cos + v — 1 . sin 0) for the form of the root, and making 

 use of Demoivre's Theorem in the expansion of each power, the possibility of satisfying the 

 equation depends upon the simultaneously satisfying the two equations, 



p^ . COS89 +p.p'' . cos 70 + q. p^ .cos 69 + r .p'' .cos 59 +s . p*. cos 40 + t. p^ .cos 39 



+ v. p^ .C08 29 + w .p.cos9 + s = 0. 



p^ .sin89+p.p~, sin 79 + q . p^ . sin 69 + r . p^ . sin 59 + s . p* . sin 4,9 + t . p^ . sin 3$ 



+ v .p* .sin29 + w.p.sin9 = 0. 



And the object of my former Memoir was, to shew that these two equations can, certainly, 

 be simultaneously satisfied. 



19. In the introductory part of this process, a principle is involved to which, on logical 

 grounds, I offer an absolute objection. I have not the smallest confidence in any result 

 which is essentially obtained by the use of imaginary symbols. I am very glad to use them 

 as conveniently indicating a conclusion which it may afterwards be possible to obtain by 

 strictly logical methods : but, until these logical methods shall have been discovered, I regard 

 the result as requiring further demonstration. It is my object in this paper to give the 

 logical certainty to which I allude, to the theorem of the roots of equations. 



20. Divested of the idea of imaginary roots, the theorem to be proved is this: "Every 

 expression, of the form of the left side of the equation given above, can be divided without 

 remainder by the quadratic trinomial x^ - 2p .cos 9 .a; + p^.'" And my process will be: 

 to effect the actual division by a" — 2p . cos 9. x + p^ ; to exhibit the form of the remainder ; 

 and to shew that the condition of evanescence of the remainder leads to the two equations 

 at the end of Article 18 (the possibility of satisfying which I consider to be demonstrated 

 in the former Memoir). 



Vol. X. Part II. 42 



