Note on the Subject of Conical Refraction. 19 



The examination of both cases is completed by the following 

 theorem : 



When three right lines at right angles to each other pass 

 through a fixed point, in such a manner that two of them are 

 confined to given planes, the plane of these two, in all its posi- 

 tions, touches the surface of a cone whose sections parallel to the 

 given planes are parabolas ; while the third right line describes 

 another cone, whose sections parallel to the same planes are 

 circles. 



The application is obvious. "We see that the curve of contact 

 in the first case is a circle. The points S in the second case are 

 also in a circle. 



NOTE ON THE ABOVE, ADDRESSED BY PROFESSOR MAC CuLLAGH TO THE 



EDITORS OF THE PHILOSOPHICAL MAGAZINE. VOL. in. 1833. 



THE introductory part of my Note which appeared in your last Number 

 was written in haste, and I have reason to think it may not be 

 rightly understood. You will therefore allow me to add a few obser- 

 vations that seem to be wanting. 



The principal thing pointed out in the Paper published some time 

 ago in the Transactions of the Royal Irish Academy is a very simple 

 relation between the tangent planes of Fresnel's Wave Surface and the 

 sections of two reciprocal ellipsoids. Now this relation depends upon 

 the axes of the sections, and therefore naturally suggested to me the 

 peculiar cases of circular sections in which every diameter is an axis. 

 Thus a new inquiry was opened to my mind. And accordingly, with- 

 out caring just then to obtain final results, which seemed to be an 

 easy matter at any time, I expressed in conversation my intention of 

 returning to the subject of Fresnel's theory in a supplementary Paper. 

 The design was interrupted, and I was prevented from attending to it 

 again, until I was told that Professor Hamilton bad discovered cusps 

 and circles of contact on the wave surface. This reminded me of the 

 cases of circular section, and the details given in my last note were 

 immediately deduced. 



c2 



