that every Algebraic Equation has a Root. 437 



viz. that it consists of a branch or branches, no one of which is 

 a courhe pointilUe, or a branch terminating abruptly in a point; 

 an algebraic cone will be in the like sense a continuous surface! 

 An algebraic curve cannot be an indefinite spiral, for in that case 

 there would be lines meeting it in an infinity of points ; and in 

 like manner an algebraic cone cannot be an indefinite spiral sur- 

 face : an algebraic cone consists, therefore, of a closed sheet or 

 sheets. An algebraic curve may indeed have conjugate or iso- 

 lated points, and an algebraic cone have conjugate or isolated 

 lines : this is a circumstance which will be adverted to in the 

 sequel. It will fix the ideas as to the general form of an alge- 

 braic cone, to remark that it may comprise twin-pair sheets, 

 such as the sheet of a cone of the second order (this is properly 

 spoken of as a twin-pair sheet, each of the two opposite portions 

 of It being called, for distinction, a twin-sheet) ; and of single 

 sheets, such as one at least of the sheets of a cone of the third 

 or any other odd order (see the annexed " Note upon Cones of 

 the Third Order ") . The advantage of the consideration of cones 

 instead of plane curves, is that we have only closed sheets, and 

 thus get rid of the distinction which exists for plane curves be- 

 tween infinite branches and the branches which are closed 

 curves. 



My proof depends on the following lemma, viz. " Consider 

 two algebraic cones with the same vertex, each of them of the 

 order n ; then if there be some one plane meeting the first cone 

 in n lines, and the second cone in n lines, such that the lines of 

 each set occur alternately, the two cones intersect in at least n 

 hnes." 



The truth of this lemma is, I conceive, a matter of intuition, 

 depending only on the notion of the continuity of the sheets of 

 the surface. Thus, if we have in piano, through a point 0, the 

 lines A, A! and B, B', such that. A, « being opposite points on 

 the same line, and so for the other lines, the order round is 

 A, B, A', B', «, /3, a', yS', it is obvious that we cannot through 

 the lines A, A' draw a cone, and through the lines B, B' draw a 

 cone without making these cones intersect in at least two lines: 

 and in like manner for two sets, each of n lines. I have, iu the 

 enunciation of the lemma, said that the cones are each of them 

 of the order n ; this was necessary in order to exclude a case 

 which might otherwise have happened, viz. a line of intersection 

 of the plane with either of the cones might have been a conjugate 

 or isolated \'\\m without any sheet through it; and if this were 

 80, we could not infer the existence of the n lines of intersection 

 of the two cones. But if a plane meet an algebraic cone of the 

 wth order in n lines, no one of these can Ijc a conjugate or iso- 

 lated line; for such line is to be considered as two or more coin- 



