RECENT ADVANCES IN SCIENCE 541 



discusses the theorem of Picard and the generalisations which 

 Borel has made. The theorem relates to the values possible 

 to a function in the neighbourhood of an essential singularity 

 of an isolated kind, and all previous demonstrations have 

 been indirect. The author gives a direct proof when the 

 singular point is of finite order. This is capable of extension 

 to an infinite order by the method outlined by the author 

 in previous notes, and a short exposition of the process is given. 



E. Cotton, Bulletin de la Societe Mathematique de France, 

 xliv. (1919), pp. 69-84, sets up certain criteria for the conver- 

 gence of infinite series of the form 2 | a„ | where « n = c n x — c n , (c„) 

 being a set of complex numbers which has (1) a finite limit 

 not zero ; (2) the limit zero. 



L. Pomey, Comptes Rendus, 170 (1920), pp. 100-101, 

 obtains some interesting theorems relating to Fermat's num- 

 bers. His starting point is the theorem of Lucas, of 1878, 

 on numbers of the form 2 m — 1 . He obtains the necessary 

 and sufficient condition that a number of this form should be 

 a prime. Three distinct necessary and sufficient conditions 

 are proved, of which any one alone will suffice. It is shown 

 further that if F = 1 + 2" is a prime, every prime number 



of type 2 m — 1 , where is an odd multiple of n, is a primi- 



rib 



tive root of F. There are further theorems stated of a similar 

 nature. 



A. Cunningham, Messenger of Mathematics, xlix. (1) (191 9), 

 pp. 1-32, discusses the factorisation of (x n ^ y") -r (x-fy) etc., 

 when x-y — 1. 



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

 develops the theory of non-harmonic Fourier series. Only a 

 portion of the paper has yet appeared, and a comprehensive 

 statement of the results is not yet possible. 



Sir George Greenhill, Phil. Mag. (3) No. 227, 1919, directs 

 attention to a posthumous memoir of Clifford, indicating the 

 greater elegance of the Clifford function as against the Bessel 

 function ordinarily used. Illustrations are given in relation to 

 problems of diffraction, whirling of shafts, stability of towers, 

 vibrations of an elastic sphere, and so forth. The paper 

 contains some interesting new cases in which the problem of 

 lateral vibrations of shafts can be solved in terms of such 

 functions. 



H. J. Priestley, Proc. Lond. Math. Soc. (2) xviii. (191 9), 

 pp. 266-268, publishes a note on the values of n which cause 



the function -7- P n ~ m (x) to vanish when x=n. The paper 



contains a proof of a theorem of Carslaw, to the effect that 



