416 REPORTS ON THE STATE OP SCIENCE. 



to displace the system from this position of equilibrium : owing to the 

 instability a small chance deviation from the state of equilibrium grows 

 into a large disturbance. Probably the best-known example of this 

 kind of action is that form of Melde's experiment in which a fine string 

 is maintained in transverse vibration by connecting one of its extremities 

 with the vibrating prong of a massive tuning-fork, the direction of 

 motion of the point of attachment being parallel to the length of the 

 string. The effect of the motion is to reader the tension of the string 

 periodically variable, and the string may settle down into a state of 

 permanent and vigorous vibration whose period is double that of the 

 point of attachment. 1 



Stephenson has developed the theory so as to obtain a mechanical 

 analogy of phosphorescence. This and other optical analogies indicated 

 by Eayleigh suggest strongly that the mathematical theory of the 

 emission of light depends either upon systems of equations with an 

 infinite number of unknown quantities or on integral equations of an 

 analogous type. Hilbert's theory of quadratic forms in an infinite 

 number of variables points to the same conclusion. 



The chief difficulty from the physical point of view is the correct 

 formulation of the equations. When this has been effected the mathe- 

 matical theories may be of very great service. 



26. Singular Integral Equations. 



When the limits of integration are infinite, or the kernel possesses 

 singularities of a certain kind, the results of the ordinary theory of 

 linear integral equations no longer hold, and in particular the homo- 

 geneous integral equation of the second kind may possess an infinite 

 number of linearly independent solutions corresponding to a given 

 characteristic number A. For instance, in the case of Fourier's -integral 



CO 



f(x) = cos xtf{t)dt 



o 



if the inversion formula holds for the function /(.r) we have also 



CO 



2f 



f(x)=\cosxtf(t)di. 







Consequently the function 



*(*) =M + ,\/%(.r) 



satisfies the homogeneous integral equation 



CO 



v(.)')= ■yjcos xtd(t)dt, 







and so it follows that for the characteristic number \/- this homo- 

 1 Of. TyncUll's Sound, 3rd ed., ch. iii. 



