to the Wave Theory of Light. 47 



so that it will be sufficient to contemplate any one of them, as 

 PM, of which the breadth PM is equal to a given line I. 



The points P and M describe, in general, similar and equal 

 curves of double curvature, which may be called ring-edges, as 

 being the edges of the ring ; and if we imagine the surface of 

 refraction, carrying these curves along with it, to be shifted 

 either way, in a direction parallel to PM, through a distance equal 

 to /, it is clear that the new position of one of the ring-edges 

 will exactly coincide with the first position of the other, and that 

 therefore the curve of the latter ring-edge will be given by the 

 intersection of the two 'equal surfaces in these two positions. 

 Let U = where U is a function of x, y, z, and given quanti- 

 ties be the equation of the surface of refraction in its original 

 position ; and, the axes of co-ordinates being fixed, suppose that 

 by the shifting of the surface the co-ordinates of a point assumed 

 on it are diminished by the given lines/, g, h, which are the 

 projections of the given line / on the axes of x, y, z, respectively. 

 Then the equation of the surface in its new position will be had by 

 substituting x + /, y + g, z + h, for x, y, z, in the equation U=Q, 

 which will thus become U + F= 0, where V is the increment of 

 U produced by the substitution. These two equations com- 

 bined are equivalent to the equations U = 0, V = 0, which are 

 therefore the equations of one of the ring-edges. If the surface 

 had been shifted the opposite way, in a direction parallel to PM, 

 the intersection would have been the other ring-edge, whose 

 equations are therefore deducible from those already found, by 

 changing the signs of/, g, h. 



52. If the equation of the surface of refraction be trans- 

 formed, so that the plane of xy may coincide with the face of the 

 crystal, and the axis of z be perpendicular to it, the origin of 

 co-ordinates being at the centre 0, no change will be produced 

 in x or in y by the motion of the surface, because PM, the direc- 

 tion of the motion, is now parallel to the axis of z ; but z will be 

 diminished or increased by / ; and, accordingly, if U' = be the 

 equation of the surface in its first position, when the centre 

 is at 0, and if U' become U' + V when z becomes z + /, the 



