PROBLEMS OF MODERN SCIENCE 



type of connecting-link between Pure Mathe- 

 matics and subjects of mathematical physics such as 

 hydrodynamics, special problems in distribution or 

 motion of electricity, the diffraction of light round 

 obstacles, and acoustic vibrations to mention only 

 a few of the main subjects concerned, though 

 perhaps those which stand out most prominently 

 in this regard. Further progress in special direc- 

 tions in these subjects is often greatly retarded by 

 the lack of some asymptotic expansion. In a 

 problem of diffraction or bending of light round 

 some obstacle of definite shape, for example, in 

 order to proceed from the quite general laws 

 governing such phenomena to an actual numerical 

 determination of the amount of light received at 

 some point behind the obstacle a matter which 

 can be quite vital in such a simple case as the 

 interpretation of phenomena occurring at the focus 

 of a lens forming part of an optical instrument 

 it is often found necessary to use mathematical 

 functions of some complexity, which are not, in 

 their usual form, at all adapted for numerical 

 calculation when some quantity in them becomes 

 large, and alternative forms are needed. These 

 alternative forms are often of a peculiar type, and 

 known as asymptotic expansions. We may say 

 that such an expansion is an expression which, 

 to a certain degree of accuracy, gives the same 

 numerical values as the function which actually 

 26 



