﻿vom 29. April 1878. 309 



Darauf legte Hr. Kummer folgende Mittheilung des corre- 

 spondirenden Mitgliedes der Akademie, Hrn. Arthur Cayley in 

 Cambridge, vor: 



On a sibi-reciprocal surface. 



The question of the generation of a sibi-reciprocal surface — 

 that is, a surface the reciprocal of which is of the same order and 

 has the same singularities as the original surface — was consider- 

 ed by me in the year 1868, see Proc. London Math. Soc. t. II 

 pp. Gl — 63, where it is remarked that if a surface be considered 

 as the envelope of a quadric surface varying according to given 

 conditions, then the reciprocal surface is given as the envelope of 

 a quadric surface varying according to the reciprocal conditions; 

 whence if the conditions be sibi-reciprocal, it follows that the sur- 

 face is a sibi-reciprocal surface. And I gave as instances the sur- 

 face which is the envelope of a quadric surface touching each of 

 8 given lines; and also the surface called the „tetrahedroid" 

 which is a homographic transformation of Fresnel's Wave Surface, 

 and a particular case of the quadric surface with 16 nodes. 



The interesting surface of the order 8, recently considered 

 by Herr Kummer, Berl. Monatsb. Jan. 1878 pp. 25 — 36, is in- 

 cluded under the theory. In fact if we consider a line L, whereof 

 the six coordinates 



«>£ )C } /, g > h •> 



satisfy each of the three linear relations 



/iß -h gib -f- 7? x c + a x f + b x g -f- cji = , 

 f,a + g. 2 b + h 2 c + a 2 f -+- b 2 g + c 2 h = , 

 / 3 « -h g z b + h 3 c -+- a 3 / -+- b 3 g -\- c z h = , 



the locus of this line is a quadric surface the equation of 

 which is 



[1878] 23 



