364 SCIENCE PROGRESS 



(save one) in the neighbourhood of its singular point. Julia 

 shews that the function assumes all finite values (save one) 

 in a certain restricted neighbourhood of the singular point. 

 Sequels to the paper appear on pp. 598-600, 718-720, 812-814, 

 882-884, 990-992, 1087-1089. 



W. H. Young, Proc. London Math. Soc. (2), xviii (191 9), 

 pp. 163-200, makes important contributions to the theory of 

 Bessel series, %A r J m (k T z), where the numbers k T are the roots 

 of the equation kj' m (k) + HJ m (k) = o. These series, which 

 are of considerable importance in various branches of mathe- 

 matical physics, have been investigated by Fourier, Hankel, 

 Harnack, Dini and Hobson, the first reliable results being due 

 to Dini. The analysis and results obtained by Prof. Young 

 are on lines quite different from those of previous writers. 



W. H. Young, Proc. London Math. Soc, (2) xviii (1919), pp. 

 1 41-162, investigates the theory of Legendre series, %a n P n (x) ; 

 by a powerful method based on the introduction of " Re- 

 stricted Fourier Series," the author makes the whole fabric 

 of the theory of Legendre series repose on the theory of 

 Fourier series. 



-J. R. Airey, Proc. Royal Soc, xcvi (a),(i9I9), pp. 1-8, gives 

 some approximate formulae for Legendre functions of high 

 order, which are extensions of approximations due to H. M. 

 Macdonald and the late Lord Rayleigh. The paper contains 

 a table from which P n (cos 6) may be calculated to six places 

 of decimals, when n exceeds 20, for each degree of the quadrant. 

 The familiar P n (cos 6) is partially replaced in this paper by 

 the notation P n (9). 



Defourneaux, Comptes Rendus, 168 (1919), pp. 880-882, 969- 

 975, investigates properties of electro-spherical polynomials ; 

 an example of such a polynomial is 2 cos n 6 qua function of 

 2 cos </>. 



E. Kogbetliantz, Comptes Rendus, \6g (1919), pp. 226-228, 

 investigates an integral, considered by Angelesco in a Paris 

 thesis 1916. The integral is a generalisation of the integral 

 obtained by Poisson's method of discussing Fourier series. 



E. Kogbetliantz, Comptes Rendus, 168 (1919), pp. 992-994, 

 169 (1919), pp. 54-57, 322-324, 423-426, examines Jacobi's 

 hypergeometric polynomials, and discusses the summability 

 of the series obtained by expanding an arbitrary function in a 

 series of them. The results resemble those obtained by Chap- 

 man and others in connexion with the summability of series of 

 Legendre polynomials. A paper on the general theory of 

 summability by the methods of Riesz (by the same author) 

 appears, ibid., 168 (1919), pp. 1090-1092, and a note on the 

 summability of Fourier series, ibid., pp. 1193-1194. 



E. A. Milne, Messenger, xlviii (1919), pp. 153-159, obtains a 



