'62 Dr. G. Green on Ship- Waves, and on Waves in 

 Note on the principle of Stationary Phase. 



The meaning of: the process which we have employed to 

 evaluate the integrals in the preceding paper is illustrated in 

 its application to the diffraction integral of the type 



1= [ S A sin 6ds , = 2tt(±-?). . . (1) 



Here S may denote any wave- surface, and p the length of 

 path from an element of surface to any point P at which the 

 resultant disturbance due to the disturbances arriving from 

 all the elements of S is required. By Taylor's theorem we 

 may express 6 in the form 



... (2) 



near any element of the surface S at which the argument 

 has the value s and the phase has the value O , 



The various elements of area ds in (1) are not equally 

 effective in contributing to the resultant disturbance at P. 

 If the phase 6 varies rapidly from element to element com- 

 pared with the amplitude term A, then the various elements 

 provide contributions which are nearly equal in value but 

 are alternately positive and negative and thus give a very 

 small resultant effect. The effective elements in (1) are 

 those for which the phase is stationary for small variations 

 in 5, that is, those in the neighbourhood of which (d0fds) = O. 

 In the optical problem this is the same as (dpfds) =0, so that 

 the condition of stationary phase for the effective elements 

 is equivalent to the usual condition for a ray — that the 

 optical path is either a maximum or a minimum. Thus 

 the principle of Stationary Phase or Group-velocity in the 

 wave-theory is the equivalent of the Law of Least Path for 

 the rays, if we look at the matter from the point of view 

 of geometrical optics. 



Then again, in the neighbourhood of any value of s, say 

 ■s , at which the stationary phase condition is fulfilled, the 

 range of values of s — s which provide the main contri- 

 bution to the value of the integral, may be regarded as very 

 small, so that the corresponding values of 6 may be taken as 

 •sufficiently represented by 



