2420 



Reality, multiplicity, separation, of roots 



2420 



Algebraic equations, theorems, 2. Myasoyedov, 



A. N. [1883] (xn) Bee. Mth. (Moscou) 11 



(1883-84) 616-. 

 Budan and Fourier's theorem. Fouret, G. N. 



A. Mth. 11 (1892) 82-, 352. 

 Sudan's method of extracting roots. Nicholson, 



P. Tilloch Ph. Mg. 60 (1822) 173-. 



theorem concerning roots of an algebraic 

 equation. Genocchi, A. Tortolini A. 7 (1856) 

 462-. 



Complex roots. Canterzani, S. Verona Mm. 

 S. It. 2 (1784) 720-. 



of equations, existence, Hermite's 

 proposition, generalisation. Studnicka, F. J. 

 Casopis 26 (1897) 95- ; Fschr. Mth. (1897) 95. 



, , proof. Kinkelin, H. Mth. 



A. 1 (1869) 502-. 

 , , theorem. Kolbe, J. Wien Sb. 



67 (1873) (Ab. 2) 188-. 

 , and factors of polynomials. Wantzel, 



L. N. A. Mth. 3 (1844) 325-. 



numerical equations. Deladereere, 

 A., <& Finck, . N. A. Mth. 3 (1844) 41-. 



Conditions that an equation of degree n may 

 have only p distinct roots. Antomari, X. 

 N. A. Mth. 16 (1897) 63-. 



Continuity of roots. Rouquet, V. N. A. Mth. 

 15 (1876) 154-. 



, proof. Bonnet, 0. Bll. Sc. Mth. As. 



2 (1871) 215-. 



Derived equations. Merlieux, E. N. A. Mth. 



3 (1844) 178-. 



polynomials, property. Young, J. R. Ph. 

 Mg. 31 (1866) 369-. 



Descartes, Eolle, Fourier, Sturm, and Cauchy, 



theorems of, deduced from one principle. 



Terquem, 0. N. A. Mth. 3 (1844) 188-, 209-, 



555-, 577-. 

 Descartes's rule, proof. Tebay, S. (vi Adds.) 



Ph. Mg. 24 (1844) 24-. 

 of signs. Terquem, 0. N. A. Mth. 2 



(1843) 248-. 

 . Finck, B. N. A. Mth. 3 (1844) 



316-. 

 . . Guilmin, A. N. A. Mth. 5 (1846) 



239-, 334-. 



Descartes's rule of signs, generalisation. Gegen- 

 bauer, L. Wien Ak. Sb. 83 (1881) (Ab. 2) 321-. 



, lowering limits furnished by. 

 Andre, D. C. E. 98 (1884) 212-. 



, theorem in complete algebraic 



polynomials applied to. Jonquieres, E. de. 

 C. E. 99 (1884) 1143- . 



theorem, extension. Lucas, E. Par. S. 

 Mth. Bll. 8 (1880) 187-. 



Distribution of roots, theorems of Laguerre and 



Myasoyedov. Klark, N. I. (xn) Kazan S. 



Nt. (Ps.-Mth.) P. 1 (1883) \_(No. 8)] 11-. 

 Double roots. Guldberg, A. S. Ts. Mth. 4 



(1886) 97-; Fschr. Mth. (1887) 86. 

 Electrical determination of roots. Lucas, F. 



C. E. 106 (1888) 195-, 268-, 645-, 1072- ; 



111 (1890) 965-. 

 Equal roots. Vachette, A. N. A. Mth. 6 (1847) 



213-. 

 . Ostrogradsky, M. A. [1849 j St Pet. 



Ac. Sc. Bll. 8 (1850) 193-. 

 . Hudson, G. QJ. Mth. 18 (1882) 215-, 



327-. 

 , condition for. Salmon, G, Camb. and 



Dubl. Mth. J. 5 (1850) 159-. 

 , theory. Rouche, E. N. A. Mth. 16 



(1857) 66-. 

 , . Mansion, . Brux. S. Sc. A. 16 



1892 (Ft. 1) 54-. 

 , , and application to tangents to curves. 



Roger, E. N. A. Mth. 3 (1844) 51-. 

 Equalities among roots, conditions for existence 



of. Cayley, A. Phil. Trans. (1857) 727-. 

 Equality of a given number of roots. Gerono, G. 



C. N. A. Mth. 1 (1842) 90- ; 6 (1847) 75-, 



2 roots, condition. Joachimsthal, F. 

 Crelle J. 33 (1846) 371-. 



Equation / 2 (z) + F/ 2 (z) = has all roots imagi- 

 nary. Cesdro, E. N. A. Mth. 4 (1885) 

 321-. 



Equations with all roots imaginary. Catalan, 

 E. C. Liege S. Sc. Mm. 15 (1888) No. 1, 



224-. 



real. Gegenbauer, L. [1881] Wien 



Ak. Sb. 84 (1882) (Ab. 2) 1102-. 



. Zehfuss, G. Grunert Arch. 34 _. Kraus, L. Casopis 15 (1886) 



(1860) 400-. 63 ; Fschr. Mth (1886) 70 



-. Walton, W. QJ. Mth. 5 (1862) J* _L __' _ fcagne M. d\ C. E. 106 

 153 -- (1888) 731-. 

 Parpaite,-. N. A. Mth. 8 (1869) _ V ! . Selivanov, D. Th. Kazan S. 



. Laguerre, E. N. A. Mth. 18 



(1879) 5-. 

 . Poincare, H. C. E. 97 (1883) 



1418-. 

 . Jonquieres, E. de. Em. N. Line. 



At. 38 (1885) 55-. 

 . Borel,E. Bll. Sc. Mth. 20 (1896) 



327-. 



(Harriot's) rule of signs, demonstration. 

 Grunert, J. A. Crelle J. 2 (1827) 335-. 



rule of signs, demonstration (Gauss). 



Ps. -Mth. Bll. 9 (1900) 51-; Fschr. Mth. 



(1900) 94-. 

 , class. Clebsch, A. [1861] 



Crelle J. 62 (1863) 232-. 

 , . Biehler, C. N. A. Mth. 



19 (1880) 149- ; 6 (1887) 9-. 



, properties. Laguerre, E. 

 N. A. Mth. 19 (1880) 224-. 



, of coefficients. Florow, 

 P. S. Fschr. Mth. (1898) 70. 



, property. Sylvester, J. J. 

 Crelle J. Mth. 87 (1879) 217-. 



signs, demonstration (Gauss). Ger- 



gonne, J. D. Gergonne A. Mth. 18 (1827-28) , source. Fouret, G. C. E. 106 



352-. (1888) 1135-. 



" , extension. Laguerre, E. C. E. , types. Fouret, G. C. E. 106 



92(1881)230-. (1888)1220-. 



164 



