4420 Bessel's Functions, etc. 



transformation of origin of reference. Walker, 



G. T. Mess. Mth. 25 (1896) 76-. 

 zeros. Hurwitz, A. Mth. A. 33 (1889) 246-. 



. Macdonald, H. M. L. Mth. S. P. 29 

 (1898) 575-; 30 (1899) 165-. 



. Schafheitlin, P. Crelle J. Mth. 122 (1900) 

 299-. _ 



Bessel's and hypergeometric series, differential 

 equations for. Schafheitlin, P. Crelle J. 

 Mth. 114 (1895) 31-. 



Catalan's theorem on roots of 



generalised. Gegenbauer, L. Wien Ak. Sb. 

 94 (1887) (Ab. 2) 801-. 



Circular opening, intensity relationships of 

 diffraction phenomenon. Steiner, L. Mth. 

 Termt. Ets. 12 (1894) 44-; Mth. Nt. B. 

 Ung. 11 (1894) 362-. 



Cylindrical functions. 



Ermakof, V. Mth. A. 5 (1872) 639-. 

 Giinther, S. Arch. Mth. Ps. 56 (1874) 292-. 

 Beltrami, E. Tor. Ac. Sc. At. 16 (1880) 201-. 

 Sonine, N. J. Mth. A. 16 (1880) 1- ; 30 (1887) 



582-. 

 analogues to. Giuliani, G. G. Mt. 25 (1887) 



198-. 

 application of theorem of Jacobi's to. Schlafti, 



L. [1871] A. Mt. 5 (1871-73) 199-. 

 definite integrals with. Hankel, H. [1869] 



Mth. A. 8 (1875) 453-. 

 expansions in terms of. Nielsen, N. N. Ts. 



Mth. 9 (B) (1898) 73-; 10 (B) (1899) 73-; 



Fschr. Mth. (1898) 361- ; (1899) 374-. 

 history. Maggi, G. A. Em. E. Ac. Line. T. 



4 (1880) 259-. 

 1st kind, theorems. Marcolongo, R. Nap. 



Ed. 28 (1889) 96-. 



and 2nd kinds. Hankel, H. [1868] Mth. 

 A. 1 (1869) 467-. 



2nd kind, for small arguments. Morton, W. B. 



[1900] Nt. 63 (1900-01) 29. 

 product of two. Nielsen, N. Mth. A. 52 (1899) 



228-. 

 representation of a function of two variables 



by. Mehler, F. G. (1871] Mth. A. 5 (1872) 



135-. _ 



Differential equation of functions of elliptic 

 cylinder. Lindemann, C. L. F. Mth. A. 22 

 (1883) 117-. 



equations for series F (p, a ; x) , F (p, a, r ; x) . 

 Pochhammer, L. Mth. A. 41 (1893) 197-. 



Discontinuous integral. Gubler, E. Mth. A. 



48 (1897) 37-. 

 Expansion of potential in morphometry of 



earth's surface. Kovesligethy, R. Mth. 



Termt. Ets. 13 (1895) 167- ; Mth. Nt. B. 



Ung. 13 (1897) 365-. 

 Expansions in series of X n functions of func- 



tions of one variable, uniformity. Dim, U. 



[1873] A. Mt. 6 (1875) 216-. 

 JF-series, generalised, reduction of differential 



equation. Pochhammer, L. Crelle J. Mth. 



112 (1893) 58-. 



Hypergeometric Functions 4420 



F-series, 3rd order, differential equations. 



Pochhammer, L. Mth. A. 46 (1895) 584-. 

 , 4th order, differential equations. Poch- 

 hammer, L. Mth. A. 50 (1898) 285-. 

 Fluctuating functions, numerical calculation 



of roots. Rayleigh, J. W. Strutt (Lord). 



(xi) L. Mth. S. P. 5 (1873-74) 119-. 

 Fourier-Bessel functions. Heine, E. Crelle 



J. 69 (1868) 128-. 

 . Kapteyn, W. Par. EC. Norm. A. 



10 (1893) 91-. 

 , expansion of function in squares and 



products of. Neumann, C. Leip. B. 21 



(1869) 221-. 

 Functions arising in continued fraction for 



e * 1 , addition theorem. Gegenbauer, L. 

 Wien Ak. Sb. 85 (1882) (Ab. 2) 491-. 



from expansion of (1 -2ax + a 2 a 2 )~^~. 

 Escary, . C. E. 86 (1878) 114-, 1451-. 



"differing as little as possible from zero." 

 Tchebitchef, P. L. Liouv. J. Mth. 19 (1874) 

 319-. 



Gaussian function F(a, 1, y, x), expansion as 



continued fraction. Tiiome, W. L. Crelle 



J. 66 (1866) 322-. 

 F(a, /3, y, x), differential equation satisfied 



by. Mathieu, E. L. Liouv. J. Mth. 8 (1882) 



357-. 



quotient j^f fr~^ r > expansion as 



continued fraction. Thome, W. L. Crelle 

 J. 67 (1867) 299-. 



Gauss's polynomials X n . Catalan, E. Brux. 

 Ac. Mm. 49 (1890-93) (No. 1) 34 pp. 



series. Thomae, J. Gott. Nr. (1884) 

 493-. 



F (a, |8, y, x), functions representable by, 



theory. Riemann, B. Gott. Ab. 7 (Mth.) 



(1856-57) 3-. 

 Geometrical series, 2nd order. Hochheim, A. 



Magdeb. Nt. Vr. Jbr. u. Ab. (1886) 137- ; 



(1887) 25-. 

 Harmonics, tesseral, functions of more than 



2 variables analogous to. Hill, M. J. M. 



Camb. Ph. S. T. 13 (1883) 273-. 

 Hypergeometric differential equation. Besso, 



D. Em. E. Ac. Line. T. 7 (1883) 230. 

 equations. Besso, D. Em. E. Ac. Line. 



Ed. 3 (1894) (Sem. 2) 393-. 



equation. Humbert, G. Par. S. Mth. Bll. 

 8 (1880) 112-. 



, derivation. Pochhammer, L. Crelle J. 

 73 (1871) 85-. 



, special cases. Boyd, J. H. A. Mth. 7 



(1892-93) 145-. 



expansions, notation suitable for. Radau, 

 R. C. E. 98 (1884) 39-. 



HYPEEGEOMETEIC FUNCTIONS. 



Pincherle, S. G. Mt. 32 (1893) 209-, 375. 

 algebraic equations soluble by. Lachtin, L. K. 



[1892] Eec. Mth. (Moscou) 16 (1893) 597-; 



17 (1895) 1-; Fschr. Mth. (1893-94) 753-. 



transformation. Papperitz, E. [1885-86] 

 Mth. A. 27 (1886) 315-. 



344 



