Expansion of Bessel Functions of High Order. 697 



The experiments here described are merely a preliminary 

 to large scale experiments in actual closed circuit telegraphy 

 the writer hopes to be able to try later on. 



One drawback to the magnetic induction form of telegraphy 

 is the rapid rate at which the effect falls off with the distance. 

 In the case of true radiation at long distances the forces vary 

 inversely as the distance, but a more rapid rate of decay, 

 something between the inverse cube and inverse square, holds 

 good for the inductive effect at least at short distances. 

 Hence the use of magnetic oscillators as transmitters is never 

 likely for this reason alone to rival the electric or open 

 oscillator, but there may be circumstances under which it 

 is possible to use them with advantage. In conclusion the 

 author desires to mention that the actual measurements 

 recorded in this paper were taken bv his assistant, Mr. Gr. B. 

 Dyke, B.Sc, with the kind help of Mr. K. W. McMillan, 

 and to these gentlemen is due an acknowledgement of their 

 share in the work, in making these observations with much 

 intelligence and care. 



LXIX. The Asymptotic Expansion of Bessel Functions of High 

 Order. By J. W. Nicholsox, B.Sc, B.A., Isaac Newton 

 Student in the University of Cambridge*. 



IN certain investigations in the theory of diffraction by 

 large obstacles the author recently found it necessary 

 to obtain some approximate formulae for the Bessel functions 

 whose order is half an odd integer. The results can be 

 applied to a large number of physical problems, and in fact 

 supply the key to the solution of the majority of problems 

 connected with the bending of waves round large spheres, 

 with which little progress has hitherto been made. The 

 Bessel functions are of several types, determined by the 

 relation between their order and argument. The attention of 

 investigators has been mainly confined to the types in which 

 the order is small in comparison with the argument, which 

 may be of any magnitude. In this paper, expressions will 

 be obtained for functions of large argument, and of order 

 comparable with, but less than that argument. This special 

 problem has received little attention, and a memoir by Lorenzf 

 appears to furnish the only contribution yet made to the 



* Communicated by the Author. 



f (Euvres Scientifiques, i. p. 405 et seq. 



