418 Dr. F. Wrightson's Remarks on 



Reasoning from analogy, it would seem that there exist trans- 

 cendental equations 



± 110 ± II<£ + Up ± Up' = const. 



± n,0 ± n^ ± n> ± n ; y= const., 



or the similar equations containing q, q 1 , instead of p, p', into 

 which these are changed by means of the transcendental equa- 

 tions between p } q, p\ a}. If in these equations we write &, </>' 

 instead of 0, <j>, it would appear that the functions Up, Up', 

 n,/?, Ufp' may be eliminated, and that we should obtain equa- 

 tions such as 



± 110 ±II£ + II0' + II^ = const. 



± 11,0 ± Tirf> ± 11,0' ± n^' = const, 

 to express the relations that must exist between the parameters 

 0, <jy and 0', <f>' of the extremities of a chord of the surface 

 # 2 + 2/ 2 +* 2 +w> 9 =0, 



in order that this chord may touch the two surfaces 



k(x i -\-y^ + z^-\-w q )-\-a^ + by 2 + cz 2 + div 2 =0, 

 k'l^ + yt + zZ + w^+a^ + byZ + czZ + dw^O. 



The quantities k, k', it will be noticed, enter into the radical 

 of the integrals II#, ilpc. This is a very striking difference 

 between the present theory and the analogous theory relating to 

 conies, and leads, I think, to the inference that the theory of the 

 polygon inscribed in a conic, and the sides of which touch conies 

 intersecting the conic in the same four points, cannot be extended 

 to surfaces in such manner as one might be led to suppose from 

 the extension to surfaces of the much simpler theory of the 

 polygon inscribed in a conic, and the sides of which touch conies 

 having double contact with the conic. (See my paper " On the 

 Homographic Transformation of a surface of the second order 

 into itself.") 



The preceding investigations are obviously very incomplete ; 

 but the connexion which they point out between the geometrical 

 question and the Abelian integral involving the root of a func- 

 tion of the sixth order, may, I think, be of service in the theory 

 of these integrals. 



LXIV. Remarks on Professor Williamson's Othyle Theory. 

 * By Dr. F. Wrightson. 



To the Editors of the Philosophical Magazine and Journal. 



Gentlemen, 



I HAVE only within the last few days had an opportunity of 

 observing a communication in the September Number of 

 the Philosophical Magazine by Professor Williamson, making my 



