RECENT ADVANCES IN SCIENCE 537 



with problems of a very fundamental character, of which a 

 somewhat extended notice appears to be called for. The 

 first is by W. H. Young, on Series of Bessel Functions. In 

 modern developments of mathematical physics, no series are 

 used more extensively, and perhaps no usual series of expansions 

 in terms of prescribed functions has been so little treated 

 from the point of view of the pure mathematician. Prof. 

 Young's paper raises such series from the level which they 

 occupied previously — a level at which the investigator, in the 

 absence of definite knowledge of the necessary and sufficient 

 conditions for their validity, has in the main been guided 

 only by the intuition of the physicist— to a new level which 

 is comparable with that of Fourier series. Prof. Young, in 

 fact, has enabled us to transfer bodily a great part of the 

 theory of Fourier series directly to expansions in terms of 

 Bessel functions. Only Dini has hitherto overcome many 

 of the serious analytical difficulties attending the development 

 of this type, — in a sense in which Fourier, Hankel, and others 

 have not contributed. We refer to the difficulties involved 

 in the discussion of the region of convergence of such series. 

 Other contributors who have dealt in a logical manner with 

 special types of such series are Kneser and Hobson. 



Prof. Young had already given a new method, based on 

 a theorem in regard to Restricted Fourier series, of discussing 

 the expansion of a function f(x) in a series of prescribed functions. 

 He developed it more particularly in regard to expansions in 

 Legendre polynomials — which were especially suitable in that 

 their asymptotic values, for large values of the order, are of a 

 trigonometric type, while at the same time, being poly- 

 nomials, their treatment did not require the use of the complex 

 variable. It was clear that the power of the method was not 

 in any real sense due to these simplifications and that it was 

 capable of equally effective use in other domains. The author 

 has selected the Bessel functions for attack in this paper — 

 not, as would have been a compelling reason, because of their 

 importance in other branches of thought, but because these 

 simplifications are no longer present. 



Some of his conclusions are of very wide significance, and, 

 taken together, form almost a complete solution of all the 

 problems of convergence of any type relating to such series 

 at internal points of the interval (0, 1). For example, under 

 the single condition that the typical term of a Bessel series 

 tends to zero, the series may be shown to behave at any such 

 internal point precisely like a Fourier series. 



Very important information is contained in the paper relating 

 to another fundamental problem in the applications of such 

 series. It has usually been assumed, in such work, that if 



