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Prof. G. G. Stokes on the Foci of Lines [June 21 , 



pencil emanating from in such a direction that its axis, after refrac- 

 tion, is perpendicular to the plate. With centre describe half a wave- 

 surface, of which only one sheet, DEF, is represented in the figure to 

 avoid confusion. In a direction parallel to the surfaces of the plate draw 



Pig. 2. 



a tangent plane to the wave-surface, touching it in E. Join OE, and 

 draw EG- a normal to the plate, and produce it to cut DF in H. Then 

 OE is the course of a ray within the crystal which, after refraction, pro- 

 ceeds in a direction perpendicular to the plate, and is therefore the axis 

 of the pencil. Eet OPQ be an adjacent ray, cutting the wave-surface in 

 P, and the tangent plane, which we may suppose to coincide with the 

 second surface of the plate, in Q. Then the retardation of the wave on 

 arriving at Q, relatively to the wave at E, will be the time the ray of light 

 takes to travel from P to Q. The form of the wave after refraction will 

 depend only on the value of this retardation, regarded as a function of 

 the two coordinates which determine the position of Q on the plate. This 

 follows at once from Huyghens's principle. If we regard QE as a small 

 quantity of the first order, the retardation will be a small quantity of the 

 second order ; and in determining the foci of the refracted pencil we only 

 want to know the retardation true to this order, and we may substitute 

 for the actual retardation any quantity which bears to it a ratio that is 

 ultimately one of equality. Hence, as the wave progresses within the 

 crystal beyond DEF. we may feign it to be travelling in an ordinary 

 medium, with a velocity of propagation equal to the actual wave-velocity 

 in the direction HE normal to the p ] ate. For if from Q we conceive a nor- 

 mal QM drawn to the wave-surface, the actual wave-velocity along MQ will 

 differ from that in the direction HE by a small quantity of the first order ; 

 and since the whole distance MQ is a small quantity of the second order, 

 we may neglect the variation of wave- velocity, and treat the meilium as 

 if it were a singly refracting one, in which a wave was travelling which 

 had already, by some means, acquired the form DEF. 



