3220 Trigonometric Series 



cases. Barbette, E. Mathesis 15 (1895) 135-. 

 30+.... Clausen, 



T. Crelle J. 8 (1832) 298-. 



. 2 s- tan ^- . Catalan, E. C. Liege S. 

 Sc/Mm. 15 (1888) ^o. 1, 227-. 



. Ssin(n!;r7r). Kopcke, H. A. Hamb. 



Mth 1 . Gs. Mt. 1 (1889) 106-. 

 class. Eogel, F. Prag Sb. (1893) (Mth.-Nt.) 



No. 33, 25 pp. 

 coefficients from particular values. Bronwin, 



B. Ph. Mg. 34 (1849) 260-, (vi Adds.) 374-. 

 convergence. Lobatschewsky, N. (vi Jdds.) 



Kazan Mm. Un. (1834) cah. 2, 226 pp. 



. Holmgren, H. Liouv. J. Mth. 16 (1851) 

 186-. 



. Poincare, H. Bll. As. 1 (1884) 319-. 



, method of increasing. Cliarlier, C. V. L. 

 Bll. As. 3 (1886) 378- ; Stockh. Ofv. (1886) 

 157-. 



, --- . Poincare, H. Bll. As. 3 (1886) 

 521-. 



of cosines of multiple angles. Fuss, N. [1809] 

 St Pet. Ac. Sc. Mm. 5 (1815) 115-. 



deduced from algebraic expansions of $lj-^~ 



and ev-l. Studnicka, F. J. Casopis 20 



(1891) 173- ; Fschr. Mth. (1891) 264. 

 determination of constants. Hunyady, J. (xn) 



Mag. Tud. Ak. Ets. 5 (No. 14) (1871) 256-. 

 differentiation. Lerch, M. C. E. 119 (1894) 



725-; Par. EC. Norm. A. 12 (1895) 351- ; 



Tor. Ac. Sc. At. 35 (1900) 54-. 

 double. Arzela, C. Bologna Ac. Sc. Mm. 4 



(1894) 373-. 

 extension of theorem. Cantor, G. [1871] Acta 



Mth. 2 (1883) 336-. 

 --- . Arzela, C. Em. B. Ac. Line. Ed. 



1 (1885) 637-. 

 multiplication. Pringsheim, A. Mth. A. 26 



(1886) 157-. 



. Bourlet, C. Bll. Sc. Mth. 13 (1889) 55-. 

 with positive coefficients. Lerch, M. Prag Sb. 



(1898) (Mth.-Nt.) No. 23, 19 pp. 



series for in terms of sines or tangents of . 



Rodrigues, 0. Liouv. J. Mth. 8 (1843) 225-. 

 of sines and cosines. Le Cointe, L. A. N. A. 



Mth. 3 (1844) 518-. 

 ---- of multiple angles. Dirksen, E. H. 



As. Nr. 1 (1823) 405-, 409- ; Berl. Ab. (1827) 



85-; (1829) 169- ; Crelle J. 4 (1829) 170-. 

 theorem. Cantor, G. Crelle J. 72 (1870) 130-. 



. Appett, P. Arch. Mth. Ps. 64 (1879) 95-. 

 theorems of Cantor and Du Bois-Eeymond, 



extension to spherical functions. Neumann, 

 C. G. Leip. Mth. Ps. B. 33 (1881) 1-; 35 

 (1883) 18-; Mth. A. 22 (1883) 406-. 

 theory of convergence applicable to other series. 

 Boussinesq, J. Liouv. J. Mth. 7 (1881) 147-. 



, fundamental formula. Bonnet, O. Bll. Sc. 

 Mth. As. 3 (1879) 480-. 



in two variables. Ascoli, G. Em. E. Ac. Line. 

 Mm. 8 (1880) 263-. 



Infinite Series 3220 



Trigonometric Series, continued. 



for y ( = tan" 1 [m tan x]) , in sines of multiples 



of x. Saporetti, A. Bologna Ac. Sc. Mm. 



8 (1887) 57-. 



Triple series. Eetali, V. G. Mt. 9 (1871) 266-. 

 Unequally convergent series. Maggi, G. A. 



Mil. I. Lomb. Ed. 26 (1893) 368-. 

 Utility of series in finding fluent. Young, 



(Dr) T. Nicholson J. 22 (1809) 213-. 

 Various series. Sarrus, F. Gergonne A. Mth. 



10 (1819-20) 217-. 

 . Tchebiche/, P. Liouv. J. Mth. 16 



(1851) 337-. 

 . Lebesgue, V. A. N. A. Mth. 18 (1859) 



433-, 460-. 



. Torelli, G. G. Mt. 10 (1872) 129-. 



. Genocchi, A. Tor. Ac. Sc. At. 10 (1875) 



985-. 



r i" ( , 1 . } . Lorgna, A. M. Verona S. It. 

 r=i \aK T * o) 



Mm. 2 (1784) 210-. 



: 2 + Abel,N.H. Crelle 



J. 1 (1826) 311-. 

 ]n_2 + 3 n ... and l n -3 n + 5 B .... Clausen, T. 



Crelle J. 7 (1831) 112-. 

 I,e x y. Malmsten, C. J. Ups. N. Acta S. Sc. 



12 (1844) 293-. 

 S (M O + MJ +...) = Sw + 2Mj + . . . , conditional 



proof. Dienger, J. Grunert Arch. 13 (1849) 



281-. 



S p n xP. Lindman, C. F. Grunert Arch. 27 

 P ~(1856) 291-. 



S, ^rv. Lebesgue, V. A. N. A. Mth. 18 

 (a + na) K 



(1859) 82-. 

 1 + T + ( H^ 2 + I S ) + ete -> calculation by 



=0 ^ ' ^ ' 



I e"e-' ze -z^ Schlomilch, 0. Schlomilch 

 Z.4(1859)433-. 



- + - + 3 + etc., a, b, c, etc. being in 

 a.b b.c c.d 



arithmetical progression. Clausen, P. A. 



Mth. Ts. 3 (1861) 69-. 



J.+J-+ etc. Frost, P. [1865] QJ. Mth. 7 

 (1866) 370-. 



Besso, D. G. Mt. 10 (1872) 160-. 



1 + _ L- +... + -_ and elliptic functions. 

 ra + 1 n + 2 2n 



Catalan, E. C. Brux. Ac. Bll. 34 (1872) 



26-, 424-. 

 SS [(x + i) (x + j)]- 1 and other series. Glaisher, 



J. W. L. [1875] Mess. Mth. 5 (1876) 21. 

 Sw n z n . Ascoli, G. Em. E. Ac. Line. At. 3 



(1876) (Pte. 2) 156-. 



1 + ... + -,sum(Catalan'sformula), 



Ti+1 n + 2 in 



generalisation. Tchebitchef, P. L. As. Fr. 



C. E. (1876) 114-. 



251 



