537] 
SOLUTIONS OF A SMITH’S PRIZE PAPER FOR 1871. 
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we consider the refraction at Q', then the index of refraction is still to be = —, that 
is, the denser medium must now be inside the small circle; the refracted ray is in the 
direction R'Q' situate symmetrically with RQ on the opposite side of the axis of y; 
and it would at first sight appear that the caustic was a curve equal and similar to 
the original caustic, but situate on the opposite side of the axis of y. But geometrically 
the complete caustic consists of two equal and similar portions situate on opposite 
sides of the axis of y; so that we really obtain, not an equal and opposite caustic, 
but in each case one and the same caustic. 
I originally obtained the theorem in a different manner; viz. the equation for the 
caustic for the first pencil of rays was found to be 
respectively.—See my “ Memoir on Caustics,” Phil. Trans., t. cxlvii. (1857), [145], p. 285.] 
5. Given at each point of space the direction-cosines (a, /3, y) of a line through that 
point: it is required to find the conditions in or'der that the lines may he not a triple hut 
a double system. 
For any given point P the values of the quantities a, /3, y which determine the 
direction of the line through that point are given as functions of the coordinates 
(x, y, z) of the point P. Hence passing from a point P to a consecutive point P' 
on the line, the coordinates of P' will be x + pa, y + p/3, z + py ; and the values of a, (3, y 
for the point P' will be 
But if the lines form a double system, we must have the same line for the point P, 
and for any other point P' on the line ; and in particular the same line for the point 
P, and for the consecutive point P'. Hence as conditions for the double system we 
obtain