420 REPORTS ON THE STATE OF SCIENCE, 



CO 



" x)} -/m 



a - \)» 



"=■ CO 



is quite noteworthy. Sufficient conditions for its validity are given by 

 Hardy. 



27. Miscellaneous Physical Applications. 



In a short abstract of a memoir which does not appear to have boen 

 printed Eouche ' indicates a number of problems in electro-magnetism, 

 mechanics, and viscosity which can be solved by means of integral 

 equations ; but none of these problems appear to have been dealt with 

 by subsequent writers. 



In 1906 Fredholm gave a theory of the lines in the spectrum which 

 promises to have interesting physical applications. 



Starting out with the idea of finding systems analogous to those 

 considered by Rayleigh and Bitz where the fundamental vibrations 

 obey laws of the same general type as the vibrations which give rise to 

 the spectra of hydrogen and other elements, Fredholm considers a 

 finite region of space over which matter is spread continuously, and 

 denotes by iv(t,x,y,z) the displacement of a particle from its position of 

 equilibrium at time t, supposing for simplicity that each particle only 

 possesses one degree of freedom. He supposes that the force F exerted 

 by one particle on another is given by an expression of the form 



F = $(£,»!,£ ; x,y,z) [>>(£, ?,;) - w(x,y,z)] . . (1) 



where $ is a symmetric function of (£,*/, £) ; (x,y,z), so that action and 

 reaction may be equal and opposite, He is thus led to an equation of 

 motion of the form 



and the existence of a fundamental vibration of the type 



w = e ixt u(x,y,z) 



depends upon the possibility of satisfying the homogeneous integral 

 equation 



u(x,y,z) 'I'(£,'/,;; x,y,z)d£,dnd<; - X 2 1 



= |||cI>(tV/,£; x,y,z)u(£,ri,Z)d!i(hd£ ... (2) 

 In order to simplify the analysis Fredholm assumes that 



*(£,jj,£; 3B,y,s)dZdri<lZ = a '. (3) 



where a is constant. Tire integral equation then becomes 



1 Comptes rendus, 1&Q0, Paris, t. 51. 



