387] 
LONDON MATHEMATICAL SOCIETY. 
21 
March 26, 1868. pp. 61—63. 
Prof. Cayley made some remarks on a mode of 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. 
If a surface be considered as the envelope of a plane varying according to given 
conditions, this is a mode of generation which is essentially not sibi-reciprocal; the 
reciprocal surface is given as the locus of a point varying according to the reciprocal 
conditions. But 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; and if 
the conditions be sibi-reciprocal, it follows that the surface is a sibi-reciprocal surface. 
For instance, considering the surface which is the envelope of a quadric surface 
touching each of 8 given lines; the reciprocal surface is here the envelope of a quadric 
surface touching each of 8 given lines; that is, the surface is sibi-reciprocal. So 
again, when a quadric surface is subjected to the condition that 4 given points shall 
be in regard thereto a conjugate system, this is equivalent to the condition that 4 
given planes shall be in regard thereto a conjugate system—or the condition is sibi- 
reciprocal ; analytically the quadric surface ax 2 + by- + cz 2 + dw 2 = 0 is a quadric surface 
subjected to a sibi-reciprocal system of six conditions. Impose on the quadric surface 
two more sibi-reciprocal conditions,—for instance, that it shall pass through a given 
point and touch a given plane,—the envelope of the quadric will be a sibi-reciprocal 
surface. It was noticed that in this case the envelope was a surface of the order 
(= class) 12, and having (besides other singularities) the singularities of a conical point 
with a tangent cone of the class 3, and of a curve of plane contact of the order 3. 
In the foregoing instances the number of conditions imposed upon the quadric surface 
is 8; but it may be 7, or even a smaller number. An instance was given of the 
case of 7 conditions, viz.,—the quadric surface is taken to be ax 2 + by- + cz- + dw- — 0 
(6 conditions) with a relation of the form 
Abe + Bca + Gab + Fad + Gbd + Hcd = 0 
between the coefficients (1 condition); this last condition is at once seen to be sibi- 
reciprocal ; and the envelope is consequently a sibi-reciprocal surface—viz., it is a 
surface of the order (= class) 4, with 16 conical points and 16 conics of plane contact. 
It is the surface called by Prof. Cayley the “ tetrahedroid,” (see his paper “Sur la 
surface des ondes,” Liouv. tom. XL (1846), pp. 291—296 [47]), being in fact a homo 
graphic transformation of Fresnel’s Wave Surface. 
{Prof. Cayley adds an observation which has since occurred to him. If the quadric 
surface ac& + by 2 + cz 2 + dw 2 = 0, be subjected to touch a given line, this imposes on 
the coefficients a, b, c, d, a relation of the above form, viz., the relation is 
A 2 bc + B’-ca + G-ab + F-ad + G-bd + li-cd = 0; 
where A, B. G, F, G, H are the “six coordinates” of the given line, and satisfy 
therefore the relation AF+BG+CH=0. It is easy to see that there are 8 lines for 
which the squared coordinates have the same values A 2 , B 2 , G 2 , F 2 , G 2 , H 2 ; these 
8 lines are symmetrically situate in regard to the tetrahedron of coordinates, and