3.52 Mr. GOODWIN, ON THE GEOMETRICAL REPRESENTATION 



It may be observed that the preceding investigation applies to multiple points in the 

 real plane by making A = 0. 



12. A less general application of what has preceded presents itself in the case of an equation 

 of an even degree having its last term positive : in this case it is well known that there is some 

 difficulty in proving the existence of a root. But I observe that if z = f{x), where /(>r) is of 

 even dimensions, sr has necessarily a minimum value, and from the minimum point an imaginary 

 branch starts off on each side of the real plane, which will stretch out to negative infinity and 

 therefore cut the plane of xy in two points which will correspond to imaginary roots. Hence 

 we see as it were the rationale of such an equation having at least two roots, for /(*') must admit 

 of a minimum, and if this be negative the curve cuts the axis of w twice, if positive imaginary 

 branches go off from the minimum and these take us down to the plane of jcy. 



13. The roots which are thus determined by the intersection with the plane of xy of 

 imaginary branches starting from points of the real curve for which /'(i) = are so related to the 

 real roots, that it has seemed to me to be desirable to denote them by a distinct name ; I therefore, 

 for want of a better name, call such roots connected roots, and those which are determined by the 

 intersection with the plane of xy of other infinite bi-anches which, as I have shewn, never cross the 

 real plane, I call isolated roots. Thus I should say of an equation of even dimensions, that it must 

 have two roots either real or connected. 



14. But more generally we may distribute the n roots of an equation into real connected 

 and isolated roots. For suppose the real branch of the curve traced, and suppose that it has. p 

 points for which f'{v) = and /'{x) does not vanish, then it is easy to see from what has been 

 said that tiiere will he p + 1 roots either real or connected; from the p maxima and minima there 

 go off 2p infinite branches which occupy 2p out of the 2m - 2 asymptotes*, leaving 2w - 2p - 2 

 asymptotes ; between each pair of asymptotes there is an infinite branch which cutting the 

 plane of xy gives a root, therefore there are n - p - 1 isolated roots; and thus we make up the 

 whole number of roots n. I will just observe that n - p - I is obviously even, because if n is 

 even p is necessarily odd, and vice versa. The same proposition may be extended to the case in 

 which other derived functions besides /'(r) vanish at any point, by the reasoning used in Art. (8) : 

 for we may consider such a point to be the degeneration of a number of contiguous maxima and 

 minima, for each of which the proposition is true. It may therefore be stated generally, that if 

 there are p real values of ,r, whether all unequal or not, which make f'{<v) vanish, then the 

 equation f(v) = has p + 1 roots either real or connected. 



15. It may be observed, that a pair of connected roots may be changed into a pair of real 

 ones by altering the position of the plane of xy, or speaking algebraically by changing the value 

 of the last term of the equation ; and this fact points out the propriety of distinguishing between 

 connected and isolated roots, which latter are necessarily imaginary wherever the plane of xy cuts 

 tlie axis of z, since they are determined by the intersection of that plane with branches of the 

 curve, which, as we have seen, never cross the real plane. 



16. The number of real and connected roots evidently depends upon the number of real 

 roots of tiie equation f'(x) = 0, and (as has been already in fact proved) if the number of real 

 roots of this derived equation be p, then the number of real and connected roots of the original 

 equation will be p + 1 ; consequently the number of isolated roots of the original equation is equal 

 to the number of imaginary roots of the derived. 



* In Art. (o) I have spoken of h - 1 asymptote?, here of 2h — 2 ; | asymptote, here for con\enience I have considered the same line as 

 the dift'erence consists merely in this, that in the former case I have Iwo stretching out to infinity on opposite sides of the origin, 

 considered the indefinite straight line through the origin as one 



