[502 
503] 
99 
we have for the 
?he condition for 
3 form function h 
itomic isothermal 
r into squares by 
:h surface of the 
3, that a quadric 
s. The analytical 
h, k of a curve 
c + &)J ’ 
quadric surfaces; 
property appears 
mimed. 
503. 
ON THE SURFACES EACH THE LOCUS OF THE VERTEX OF A 
CONE WHICH PASSES THROUGH m GIVEN POINTS AND 
TOUCHES 6—m GIVEN LINES. 
[From the Proceedings of the London Mathematical Society, vol. iv. (1871—1873), 
pp. 11—47. Read January 11, 1872.] 
I consider the surfaces, each of them the locus of the vertex of a (quadri-)cone 
which passes through m given points and touches 6 — m given lines; viz. calling the 
given points a, b, c,... and the given lines a, /3, 7,..., the surfaces in question are: 
abcdef 
Order 
4 
abode a. 
8 
abcda/3 
16 
abca/37 
24 
aba/3 yS 
24 
aafiyS e 
14 
aj3y8e £ 
8 
I remark that the orders of these several surfaces are in effect determined by 
the investigations of M. Chasles in regard to the conics in space which satisfy seven 
conditions. The surface cibcdef was long ago considered by M. Chasles, and it is treated 
of in my “ Memoir on Quartic Surfaces,” [445], and in the same Memoir the surface 
a/3y8e£ is also referred to: these two surfaces, and also the surfaces aa/3ySe and aba/3<yB 
are considered by Dr Hierholzer (*) in his excellent paper “ Ueber Kegelschnitte im 
Raume,” Math, Annalen, t. 11. (1870), pp. 563—586, and to him are due the equations 
given in the sequel for the surfaces abcdef and a/3y8e£: the researches of the present 
Memoir are in fact a continuation and development of those in the Memoir last 
referred to. 
1 I was grieved to hear of Dr Hierholzer’s death last autumn, at Carlsruhe, at the early age of 30. 
13—2