2420 



Reality, multiplicity, separation, of roots 



Equations, difference product in case of equal 

 roots. Franke, E. [1876] Crelle J. Mth. 83 

 (1877) 65-. 



with a given number of complex roots. 

 Gegenbauer, L. [1882] Wien Ak. Sb. 87 

 (1883) (Ab. 2) 264-. 



no positive root, conditions. Andre, D. 



C. E. 98 (1884) 561-. 

 , 3rd & 4th orders, nature of roots, criteria. 



Gaines, R. E. A. Mth. 1 (1900) 111-. 

 Fourier and Sturm, theorems. Crelle, A. L. 



Crelle J. 13 (1835) 119-. 



, , graphic representation. Tanner, 



.H.W.L. Mess. Mth. 18 (1889) 95-. 

 , theorem on roots of equations and re- 

 curring series. Stern, M. A. Crelle J. 9 



(1832) 305-. 

 Fourier's rule, proof. Young, J. R. Ph. Mg. 



23 (1843) 6-. 

 for real roots. Hill, C. J. D. Lund 



Acta Un. 8 (1871) (Mth.) No. 2, 8 pp. 



theorem on equations. O'Kinealy, J. Ph. 

 Mg. 48 (1874) 95-. 



. Sharp, W. J. C. QJ. Mth. 17 



(1881) 132-. 

 Functions whose roots separate each other. 



Markow, W. A. Fschr. Mth. (1899) 100. 



. Iwanow, J. J. Fschr. 

 Mth. (1899) 100. 



. Markow, W. A. Fschr. 



Mth. (1899) 100- . 

 Imaginary and equal roots, criteria. Homer, J. 



[1867] QJ. Mth. 9 (1868) 57-. 



roots. Cauchy, A. L. , Par. S. Phlm. Bll. 



2420 



Imaginary roots, Newton's rule. Sylvester, 

 J. J. [1865] L. Mth. S. P. 1 (1866) No. 2, 

 16 pp. ; Ir. Ac. T. 24 (1871) 247-. 



, , invariantive criteria for quintic. 



Sylvester, J. J. Phil. Trans. 154 (1864) 

 579-. 



, for number. Lobatto, B. Amst. 



Vs. Ak. 9 (1859) 92-. 



, . Minich, S. R. Yen. At. 



11 (1865-66) 127-. 



, , proof. Young, J. R. Ph. 



Mg. 30 (1865) 113-, 289-, 363-; 31 (1866) 

 369-; 32 (1866) 121- ; Ir. Ac. P. 10 (1870) 

 343-. 



, , Sylvester's theorem. 

 Poulain, A. Les Mondes 11 (1866) 16-. 



, , . Genocchi, A. N. 



A. Mth. 6 (1867) 5-. 



, and Sylvester's theorem. Jon- 

 quieres, E. de. C. B. 99 (1884) 62-, 111-, 

 165-, 269-. 



, , theorem connected with. Syl- 

 vester, J. J. [1879] Mess. Mth. 9 (1880) 



, number. Berard, . Gergonne A. 

 Mth. 9 (1818-19) 345-. 



, between limits. Dienger, J. Grunert 



Arch. 21 (1853) 361-. 

 , , test for. Lobatto, R. Liouv. J. 



Mth. 9 (1844) 295-. 

 of numerical equations, number. Anon. 



(vi 529) Gergonne A. Mth. 9 (1818-19) 60-. 

 , test for. Lavernede, T. Gard. 



Not. Tr. Ac. (1809) 195-. 



(1817) 5-, 161- ; Par. EC. Pol. J. 11(1820) , . Faade Bruno, F. Liouv. 



(18e cah.) 411-. J. Mth. 15 (1850) 363-. 



, Eolle's theorem extended to. Liou- 

 ville, J. Liouv. J. Mth. 9 (1864) 84-. 



Brux. S. 



. Bonnet, O. N. A. Mth. 4 (1845) 236-. 



. Terquem, 0. N. A. Mth. 5 (1846) 141-. 



. Prouhet, E. N. A. Mth. 14 (1855) 199-. 



. Buys-Ballot, C. H. D. Amst. Vs. Ak. 



7 (1858) 316-. 



. Heaivood,P.J. Mth. Gz. No. 4 (1895) 31-. 



. Vaes, F. J. N. Arch. Wisk. 3 (1898) 117-. 



, Budan's criterion. Christie, J. R. Ph. 



Mg. 21 (1842) 96-. 



, Cauchy's theorem, extension. De Mor- 

 gan, A. [1857] (DC) Camb. Ph. S. P. 1 

 (1866) 194. 



, , proof. Liouville, J., & Sturm, C. 



Liouv. J. Mth. 1 (1836) 278-. 



, , proofs. Stiirm, J. C. F. Liouv. 



J. Mth. 1 (1836) 290-. 



, criteria. Young, J. R. Ph. Mg. 22 

 (1843) 186-, 252-; 23 (1843) 450- ; 29 (1846) 

 32-. 



, . Colombier, P. A. G. N. A. Mth. 



7 (1868) 308-, 501-. 

 of equations of order higher than 4th. 



Carandino, G. G. Arcad. 32 (1826) 90-. 



, existence. Dupre, A. Gergonne A. 

 Mth. 18 (1827-28) 68-. 



, . Gergonne, J. D. Gergonne A. Mth. 



19 (1828-29) 124-. 

 , _, test. Serret, P. N. A. Mth. 6 



(1847) 200-. 

 , limiting values of coefficients for given 



number. Schramm, H. 



(1866) 325-. 



, . Mansion, . 

 Sc. A. 13 (1889) (Pt. 1) 42-. 



, rules analogous to Newton's. Minich, 



S. R. Ven. At. 11 (1865-66) 211-. 

 , for inferior limit to number. Minich, 



S. R. Ven. At. 11 (1865-66) 309-. 

 , theorem. Sturm, J. C. F. N. A. Mth. 



5 (1846) 115-. 

 Infinite roots of equations. Gerono, G. C. 



N. A. Mth. 3 (1844) 32-, 85-. 

 , and asymptotes. Gerono, G. C. 



N. A. Mth. 4 (1845) 33-. 



Integral functions Y n = II ( y - 2 cos -r |i and 

 = 1\ n + l/ 



Sturm's functions. Vries, J. de. Amst. 



Ak. Vs. 4 (1896) 133-. 

 Interlacing of functions. Sylvester, J. J. 



Crelle J. Mth. 88 (1880) 1-. 

 Laguerre's polynomials, property. Laurent, H. 



N. A. Mth. 19 (1880) 380-. 

 Least root, general expression for. Konig, J. 



Mth. A. 9 (1876) 530-. 



roots. Dickson, J. D. H. [1877] Edinb. 

 E. S. T. 28 (1879) 119-. 



Limit, inferior, of roots of an algebraic equation. 

 Mlasojedov, A. N. Eec. Mth. (Moscou) 12 

 (1885) 22-. 



Arch. Mth. Ps. 45 , , . Fouret, . Par. 



S. Mth. Bll. 20 (1892) 4-. 



165 



