[Extracted from The Quarterly Journal of Pure and Applied Ufatfiematics, No. 34, 1867.] 



XXXVII. On the Cyclide. 



l\ optical treatises, the primary and secondary foci of a small pencil are 

 sometimes represented by two straight lines cutting the axis of the pencil at 

 right angles in planes at right angles to each other. Every ray of the pencil 

 is supposed to pass through these two lines, thus forming what M. Pliiker* 

 has called a congruence of the first order. 



The system of rays, as thus defined, does not fulfil the essential condi- 

 tion of all optical pencils, that the rays shall have a common wave-sur; 

 for no surface can be drawn which shall cut all the rays of such a pencil 

 at right angles. 



Sir W. R. Hamilton has shewn, that the primary and secondary foci are 

 in general the points of contact of the ray with the surface of centres of 

 the wave-surface, which forms a double caustic surface. If we select a pencil 

 of rays corresponding to a given small area on the wave-surface, their points 

 of contact will lie on two small areas on the two sheets of the caustic 

 surface. The sections of the pencil by planes perpendicular to its axis will 

 appear, when the pencil is small enough, as two short straight lines in pi 

 perpendicular to each other. 



I propose to determine the form of the wave-surface, when one or both 

 of the so-called focal lines is really a line, and not merely the projection of 

 a small area of a curved surface. 



Let us first determine the condition that all the normals of a surface 

 may pass through one fixed curve. 



Let R be a point on the surface, and RP a normal at P, meeting the 

 fixed curve at P. Let PT be a tangent to the fixed curve at P, and HI'T 

 a plane through R and PT. 



* Philosophical Transactions, 1864. 



