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[671 
671. 
ON A SIBI-RECIPROCAL SURFACE. 
[From the Berlin. Akad. Monatsber., (1878), pp. 309—313.] 
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 considered by me in the year 1868, see Proc. London Math. Soc. t. n. 
pp. 61—63, [part of 387], 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 surface is a 
sibi-reciprocal surface. And I gave as instances the surface 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 quartic surface with 16 nodes. 
The interesting surface of the order 8, recently considered by Herr Kummer, Berl. 
Monatsber., Jan. 1878, pp. 25—36, is included under the theory. In fact, if we consider 
a line L, whereof the six coordinates 
a, b, c, f g, h, 
satisfy each of the three linear relations 
f x a + gjb 4- Kc + Oi/+ Kg + cji = 0, 
f/i + gj) + h,,c + a 2 f+ h,g 4- cji = 0, 
f 3 a + g jo 4- he 4- a 3 f+ b 3 g 4- c 3 h = 0, 
the locus of this line is a quadric surface the equation of which is 
T = (agh) x 2 4- (bhf) y 2 4- (cfg) z 2 4- (abc) w 2 
+ [(<%) - (cah)] xw + [(b/g) 4- (chf)] yz 
+ [( b ch) - (abf)] yw + [(cgh) + (afg)] zx 
+ [(caf) - (beg) ] zw + [(ahf) 4- (6^)] xy = 0,