THE 



341 



THE 



extended algebra, Z=x+y V-l. Again, let ft and v be 

 the co-ordinates of the radical point, and A its radius vec- 

 tor; so that A = p + w'-l. Let R be the radius drawn 

 from the radical point to the contour, so that Z = A -)- R, 

 R being infinitely small. By hypothesis p+vj-l is 

 a root of <f>s = ; let there be m equal roots belonging 

 to the radical point (m being 1, or some other integer) : 

 then will e(A + R) be capable of expansion into the 



form BR + B^+'+.&c., of which, R being infinitely 

 small, only the first term need be considered. Now let 

 B and R (taking the most general forms) be b (cos /3 



+ sin/3 . V - 1) and r (cosp + sin p V 1), whence BR 

 will be 



br m { cos(mp-t-/3)+sin(ff!p+/3). */-l}' 

 and P : Q will be cot (mp+/3), its remaining terms being 

 infinitely small. Let R make a complete circuit, or let p 

 increase from to 2ir, whence mp -f- will go m times 

 through four right angles. In each revolution cot (mp+/3) 

 will change from + to - twice, passing through nothing : 

 but never from to 4- except by passing through infinity. 

 The theorem is then true : for k = 2i, 1 = 0, $(k l) = m, 

 and there are m radical points (or one radical point be- 

 longing to m equal roots) within the contour. 



The theorem is then true for every infinitely small con- 

 tour. Next, let the whole contour ABCD be divided into 

 an infinite number of infinitely small figures, with no 

 other limitation than that no radical point is to fall upon 

 one of the lines of division. Let a point move round 

 each of the infinitely small figures in the positive direc- 

 tion of revolution. It is clear that the expression $(2A 

 2/1 will not be altered if we remove all the internal 

 division lines and leave only the external contour ABCD : 

 for each internal line is described by two points moving 

 in opposite directions, and wherever one point adds a unit 

 to 2A, the other adds one to 2/. Hence the value of 

 2/{ / can be found by finding that of k I for the boun- 

 dary only : and the theorem is proved. 



If <j>Z = AZ + A,Z + , and if we make the con- 

 tour in question a circle with the origin as a centre, and 



a radius so great that the highest term AZ" need be the 

 only one retained, we can immediately prove that <Z has 

 neither more nor less than n roots. Jor, Z being z (cos ? 

 4- sin ? V 1 ) and A being a (cos n + sin a . / 1)> we 

 find as before that P : Q, or all of it that need be con- 

 sidered, is cot (n+a), whence k = 2n, 1=0, and -J (kl) 

 = n. 



4. We may now refer to STURM'S THEOREM, to Fourier's 

 theorem (given in the article just cited), to Des Cartes' 

 theorem, a very limited particular case of Fourier's, and to 

 Homer's adaptation of, and addition to, the old method of 

 numerical solution by Vieta (an account of the history of 

 this last problem is given in the ' Companion to the 

 Almanac' for 1839). We have then, since the beginning 

 oi this century, a complete theoretical mode of determin- 



'!ie number of roots, real or imaginary, between any 

 given limits ; both exceedingly difficult in the complica- 

 tion of the operations which they require. Also, a mode 

 of easy application, though not theoretically perfect, of 

 determining the limits between which the real roots lie ; 

 and a process for the numerical solution which places that 

 question upon the same footing as the common extraction 

 of square, cube, &c. roots ; making those extractions 

 themselves, except only in the case of the square root, 

 much easier than before. 



5. The Newtonian method of approximation is in the 

 following theorem. If a be nearly a root of <jix=0, and 

 if <j>a : (f>'a be small, then 



ipa 



~ 0^ 



is more nearly a root. See APPROXIMATION for the use of 

 this, and TAYLOR'S THEOREM, p. 129, for a more extensive 

 result. But the use of Horner's method is very much 

 more easy than that of Newton : the former, in fact, in- 

 cludes and systematizes the latter. But this remark 

 applies only to algebraical equations : for all others New- 

 t'orm just given remains practically unamended. 



6. We refer to the article ROOT for the solution of 



x 1 = 0. The following equation, x " 2 cos . x 



+ 1 = 0, admits of complete solution on the same princi- 

 ples. 



7. If <pa and 06 have different signs, one or some other 

 odd number of roots of <?x lies between a and b but if 

 they have the same signs, either no one or an even number 



)t roots lies between a and b. Every equation of an odd 

 degree has at least one real root, negative or positive ac- 

 cording as the first and last terms have like or unlike 

 signs. Every equation of an even degree having the first 

 and last terms of unlike si-gns has aUeast two real roots 

 one positive and one negative. 



8. If all the coefficients of $x be real, and one of the 

 two, a&V-l, be a root, so is the other: and if all the 

 coefficients be rational, and one of the two, a ,76, a and b 

 being rational, be a root, so is the other. If there be a 

 rational fractional root, its denominator must be a divisor 

 of the first coefficient, and its numerator of the last, as 

 soon as the equation p=0 is cleared of fractions. N.B. 

 Among the divisors of a number we reckon 1 and itself. 



9. In the equation a a x"+a, x*~ l +a, x"~ 2 + ... + a 

 x+a n = 0, the sum of all the roots is -a, : n a , the sum 

 of the products of every two is a t :a , that of the products 

 of every three is -o 3 : a , and so on. Finally, the pro- 

 duct of all the roots is a^ : a , according as n is even or 

 odd. And if r, , r,, ... r^ be the roots, then a x*+ ... is 

 the same as a (a; r,} (,T r 2 ) .... (xr ). 



10. If the preceding expression be called Ax, and 



B 1 n o 



na a x + (n l)a l x '+..., its derived function, be 

 called <J>'x, we have 



(fix x r t xr, ' ' xr ' 



and if tyx be any rational and integral algebraical function 

 of a-, the sum ij/r, + i//r 2 + .... + ^r n is the coefficient of 



the highest power of x in the remainder of the division of 

 "x X ^/x by <jix. 



11. If S n in all cases stand for the sum of the th 

 powers of the roots of the equation, we have 



S = n, ff S, + a, = 0, o S 2 + a, S, + 2a t = 



and so on up to 



*" C* i Q i i n 



' U i O?j_ 1 f" dn ^5n 2 ~T "T" ??(Z ~~ (J 



after which, in all cases, 



Hence also the coefficients of the expression may be 

 found in terms of S, S, .... S n> us soon as </ is given. 



12. All rational symmetrical functions of the roots may 

 be easily expressed in terms of S, S 2 , &c., and thence in 

 terms of the coefficients of the expression. 



13. If it be required to find a function tyy the roots of 

 which shall be given functions of those of Ax, so that in 

 all cases y = Far, proceed as in finding the highest com- 

 mon divisor of <.r and Fxy, and take for $y the final 

 remainder. But if this final remainder should be of a 

 higher dimension than, from- the known number of its 

 roots, it ought to be, it will be a sign that some of the 

 factors introduced in tne process have affected the re- 

 mainder, and these must be examined and removed. The 

 treatment of this case belongs to the general question of 

 elimination, but the following particular cases are almost 

 all that are necessary. 



14. To decrease all the roots of Ax by a given quantity, 

 or to make y=xa, or x=y+a, observe that the result- 

 ing equation must be 



A*'& A d _ 



' .A...A 



where the coefficients Aa, A'a, J A"a, &c. may be most 

 readily found by the process described in INVOLUTION 

 (p. 7). The same process maybe applied, by using a 

 instead of a, to increase all the roots of <f>x by a given 

 quantity. It is by this process that the second term of 

 an equation is taken away: thus, the equation being 



a <t x n +a i x"~ l + ... = 0, assume 



