436 Mr, A. Cayley on a Proof of the Theorem 



discharged from the blood iu greater quantity than it had been 

 supplied to that fluid, and the supply had become exhausted ; 

 but this diminution and cessation in the increase may not indi- 

 cate any defect of carbonic acid still in the blood, but rather the 

 cessation of that action of the food which causes the increased 

 elimination ; for whenever a new supply of food is given, there 

 is a renewal of the increased discharge of carbonic acid. 



LXVII. Sketch of a Proof of the Theorem that every Algebraic 

 Equation has a Root. By A. Cayley, Esq.^ 



T HAVE referred to the theorem as usually stated ; for it is 

 -■- an easy consequence of the existence of a single root of an 

 equation of any order, that for an equation of the nth order there 

 are n roots : the proof here proposed goes, however, to show 

 directly the existence of the n roots : it is in form a geometrical 

 one, and was suggested to me some months ago by a letter from 

 Prof. De Morgan, containing the remark made in his memoir, 

 " A Proof of the Existence of a Root in every Algebraic Equa- 

 tion," &c. (Camb. Phil. Trans, vol. x. 1858), viz, " that the curves 

 P=0, Q = 0, the intersections whereof determine the root-points, 

 ai'e such that two branches, one of each curve, cannot enclose a 

 space." The proof which occurred to me was in character some- 

 what similar to that given by the Astronomer Royal in the paper, 

 " Suggestion of a Proof of the Theorem that every Algebraic 

 Equation has a Root " (Camb. Phil. Trans, vol. x. 1858), and 

 which was suggested to him by Prof. De Morgan's memoir. I 

 have since varied my proof by considering therein cones in the 

 place of plane curves. It will be obvious, upon reading it, that 

 the proof is closely connected with Cauchy's well-known theorem 

 for the number of roots within a given circuit ; the circuit being 

 in this case infinity, and the number of roots included within it 

 consequently equal to the order of the equation. 



The curve repi'csented by an equation of the nth degree be- 

 tween the coordinates [x, y) is by definition a curve of the wth 

 order ; and a cone standing on any such curve (taking the ver- 

 tex for origin) is represented by a homogeneous equation of the 

 nth degree between the coordinates {x, y, z), and is by definition 

 a cone of the nth order. It is very easy to show that an equa- 

 tion of the nth degree cannot have moi'e than n roots ; and we 

 have thence the geometrical theorems, that a curve of the nth 

 order is not intersected by a line in more than n points, and that 

 a cone of the nth order is not intersected by a plane (I speak 

 throughout of planes through the vertex) in more than n lines. 

 I assume that an algebraic curve is always a continuous curve, 

 * Communicated by the Author. 



