120 Mr. R. F. Gwyther on 



co-ordinates only. The method is not quite the same as is 

 developed in the rest of the paper, but it will, I hope, help 

 to explain the more complex portion which follows. 

 Except in connecting expressions on the basis of per- 

 manence of form, the paper contains no original results, 

 unless it is in the introductory example. 



i. As a first example of the condition of permanency of 

 form of the expressions for a physical quantity in a con- 

 tinuous medium, I take one which is simple — in the sense 

 that it deals with the mode in which the co-ordinates them- 

 selves enter the expression considered (whereas at a later 

 stage the arguments will be functions of the co-ordinates) — 

 and which is also definite in its statement, while later I 

 shall consider quantities generically ; — 'namely, the case of a 

 plane polarised wave of light falling perpendicularly on and 

 diffracted by a circular aperture of any size. 



Consider the wave to travel in the positive direction 

 along the axis of x, take the centre of the circular aperture 

 as the origin and as axes of y and z two lines at right 

 angles in the plane of the aperture, and let the displacement 

 in the incident wave make an angle a with the axis of y. 



Our object is to determine, as far as the permanency of 

 form conditions will allow, the mathematical forms which 

 must be taken to represent the displacement in the secondary 

 wave on the positive side of the plane of yz. Consider the 

 displacement at a point in the secondary wave which 

 corresponds to the component cosa along the axis of y, and 

 write its components 



U =f(x, r, 0)cosa 

 Y = (p(x, r, 0)cosa 

 W = \p(x, r, 0)cosa 



where x, r and 6 are the cylindrical co-ordinates of the 

 point. 



It follows at once that the components of the displace- 



