2420 



Reality, multiplicity, separation, of roots 



2420 



Eeal roots, number between limits. Petr, K. 



Prag Ceske Ak. Fr. Jos. Kz. (Tfida 2) 6 



(1897) Art. 8, 11 pp. ; Prag Fr. Jos. Ac. Sc. 



Bll. (Mth.-Nt.) 4 (1897) (Pt. 2) 61-. 

 -- , --- , Sturm's theorem. Joachims- 



thai, F. Crelle J. 48 (1854) 386-. 

 -- , for equation x m -px+q = Q. Regis, 



D. G. Mt. 8 (1870) 226-. 



, series for number. Brioschi, F. N. 

 A. Mth. 15 (1856) 264-. 



-- , their upper and lower limits and sepa- 



ration. Steklov, V. A. Kharkov Mth. S. 



Com. 3 (1893) 103- ; Fschr. Mth. (1891) 



94-. 

 Eeality of roots. Bjorling, C. F. E. [1868] 



Stockh. Ofv. 25 (1868-69) 241- ; Arch. Mth. 



Ps. 48 (1868) 363-. 

 --- . Favero, G. B. G. Mt. 13 (1875) 



249-. 



, conditions. Griveau, A. Les 

 Mondes 17 (1868) 214-. 



--- of discriminating cubic. Gerono, G. C. 



N. A. Mth. 11 (1872) 305-. 

 ---- equation of 3rd order. Grunert, 



J. A. Grunert Arch. 29 (1857) 442-. 



in transformation theory. 

 Saalschiltz, . Konigsb. Schr. 38 (1897) 

 [17}-. 



----- ( l -^\ 

 \l-ixj 



C. A. As. Fr. C. E. (1882) 102-. 



-- important equation, new proof. 

 Gravelaar, N. L. W. A. [1875] Arch. 

 Mth. Ps. 58 (1876) 301-. 



---- quintic, invariant criteria. Harris, 



E. A. A. Mth. 5 (1889-90) 217-. 

 ----- , -- . Petr, K. Prag Ceske 



Ak. Fr. Jos. Ez. (Tfida 2) 8 (1899) Art. 23, 

 12 pp. ; Fschr. Mth. (1899) 103. 

 ---- secular equation, proof by Fourier's 

 method. Sylvester, J. J. Crelle J. Mth. 88 

 (1880) 4-. 



Laisant, 



Eoots of cubic, criteria. Vyvyan, T. G. Mth. 

 Gz. 1 (1900) 39. 



, equal numerically, of opposite sign, cri- 

 teria. Mtiller, F. Arch. Mth. Ps. 46 (1866) 

 32-. 



of an equation, limits to. Gob, A. Mathesis 



9 (1889) 269-. 



, multiplicitity of, and a limit to. 

 Demoulin, A. Mathesis 9 (1889) 268-. 



equations. Spitzer, S. Grunert Arch. 



25 (1855) 137-. 



, analysis. Murphy, E. Phil. Trans. 



(1837) 161-. 



with complex coefficients. Hermite, 



C. Crelle J. 53 (1857) 182-. 



, elementary representation and inspec- 

 tion. Bjnrling, C. F. E. Ts. Mt. Fys. 1 

 (1868) 119-. 



with integral coefficients. Cauchy, 



A.L. C. E. 24 (1847) 407-. 



, nth degree, with application to bi- 

 quadratic. Mossbrugger, L. Grunert Arch. 

 28 (1857) 205-. 



, surd. Moon, E. Camb. Mth. J. 3 



(1843) 20-. 



, theorems. Fleury, L. L. Cherb. 



Mm. S. Sc. 8 (1861) 373-. 



, new formulae relative to. Silva, J. A. M. 

 da. [1882] G. Teix. J. Sc. 4 (1883) 3-. 



, number satisfying a given condition. 

 Cauchy, A. L. G. E. 40 (1855) 1329-. 



, theory. Bourget, J. N. A. Mth. 9 (1870) 

 505-. 



_, _. Bjorling, C. F. E. Stockh. Ak. Hndl. 



10 (1871) (No. 3) 53 pp. 



Eule of signs, new theory. Meech, L. W. 



Des Moines Anal. 2 (1875) 81-, 97-. 

 Eules of signs, combinations of four. Sylvester, 



J. J. C. E. 61 (1865) 282-. 

 Separation of nearly equal roots. Christie, J. E. 



Ph. Mg. 21 (1842) 96-. 

 real roots of equation. Schramm, Ht 



Grunert Arch. 36 (1861) 420-. 



, theorem. Korteweg, D. J. N. Arch. equations. Ovejero, J. Arg. S 



Wisk. 4 (1899) 46-. 



and signs of roots. Du Bourguet, . Par. 

 S. Phlm. Bll. (1815) 14-. 



Eolle's theorem. Collin, J. N. A. Mth. 20 



(1881) 132-. 

 , and an application. Biehler, C. N. A. 



Mth. 6 (1887) 75-, 190-. 



, extension. Boer, F. de. Amst. Ak. 

 Vs. M. 19 (1884) 384-; Arch. Neerl. 19 

 (1884) 207-. 



, . McCulloch, J. F. A. Mth. 4 

 (1888) 5-. 



, generalisation. Lucas, F. C. E. 106 



(1888) 121-. 



, Lagrange's, and Cauchy's, geometri- 

 cally equivalent. Mansion, P. [1875] Mess. 

 Mth. 5 (1876) 34-. 



Boot-limitation, Cauchy's theorems, geometri- 

 cal representation. Cayley, A. [1874] 

 Camb. Ph. S. T. 12 (1879) 395-. 



Eoots, commensurable, criteria. Mathieu, E. 

 N. A. Mth. 16 (1857) 145-. 



, fractional, criteria. Terquem, 0. N. A. 

 Mth. 4 (1845) 397-. 



Ci. A. 27 (1889) 169-. 

 roots. Dirksen, E. H. Berl. Ab. (1835) 



337-. 

 . Umpfenbach, Dr. Crelle J. 20 



(1840) 60-. 

 . Transon, A. N. A. Mth. 7 (1868) 25-, 



57-. 

 . Bjorling, C. F. E. Ups. N. Acta S. 



Sc. 7 (1870) (No. 7) 35 pp. 

 . Maleyx, L. N. A. Mth. 11 (1872) 



404- ; 13 (1874) 385-. 



. Laurent, H. N. A. Mth. 14 (1875) 37-. 



. Mlasojedov, A. N. [1884] Eec. 



Mth. (Moscou) 12 (1885) 433- ; Fschr. Mth. 



(1885) 63. 

 of certain equations. Laguerre, E. 



G. E. 92 (1881) 178-. 

 , improved statement of rule, and new 



theorem. Sylvester, J. J. Ph. Mg. 31 (1866) 



214-. 

 , Laguerre 's theorem. Lucas, E. N. 



A. Mth. 19 (1880) 145-. 

 , . Levy, L. N. A. Mth. 19 



(1880) 148. 



167 



