254 MR. A. CAYLEY’S ANALYTICAL RESEARCHES CONNECTED WITH 
§ 1 . Lemma relating to the sections of a surface of the second order. 
If 
ax^-\-hy^-\- cz^ duP -\-2fyz -\-2gzx -\-2hxy -\-2lxw -\-2myw -\-2nzic=0 
be the equation of a surface of the second order, and 
^x^ -\-^y'^ -\-2^y%-\-2<Qzx -\-2\^xy -^2\xui) -\-2£^yw^2mic^Q 
the reciprocal equation, the condition that tlie two sections 
"Kx -\-\xy -\-vz -\-^w =0 
l!x -\-fy + v'z + = 0 
may touch, is 
(9[^"+B/-t-^+C^^+Df'+2jTp+2(25j'X+2^X;i0+2lL?v^+2i¥l^^^+2J13t'e)i 
X{9iK'^+^f^+€v'^+^f+2S^fv'+20fX'-^2^X'f+2^X'^' + 2£^fp’+2B’^'§')^. 
= d" J^( d“ “1- ^ (I'X^ -j- v'X) “I- 1 ^ [xf d" d“ '^i) 
d-iil(^^' d-f^'f ) d- B {v^' d- . 
And in particular if the equation of the surface be 
ax^ d" by^ d" d" ‘^fd ^ d“ -{-2hxy ■\-pw^ = 0, 
the condition of contact is 
d” 3$///^ d" d“ -\-2(BvX-\-2 ILX^ jo 
(^ix!^ d- d- d- 2 d- 2<Bv'X' -j- 2^X'f + " 
= ^9[XX' d- -\- fv) d" # (j'X' -f- d" ^ d- ^ V) f ) ’ 
in which last formula 
^—hc—f^, ^=ca—g^, ^^ah—h^, 
f=gh-af iB = hf-bg, l=/^-c/i, 
Ys.=.abc—af^—b^—ch^-\-2fgh. 
§ 2 . 
In order to state in the most simple form the geometrical construction for the 
solution of Steiner’s extension of Malfatti’s problem, let the given sections be 
called for conciseness the determinators*; any two of these sections lie in two dif- 
ferent cones, the vertices of which determine with the line of intersection of the 
planes of the determinators, two planes which may be termed bisectors ; the six 
bisectors pass three and three through four straight lines ; and it will be convenient 
to use the term bisectors to denote, not the entire system, but any three bisectors 
passing through the same line. Consider three sections, which may be termed tactors, 
each of them touching a determinator and two bisectors, and three other sections 
(which may be termed separators) each of them passing through the point of contact 
* I use the words ‘ determinators,’ &c. to denote indifferently the sections or the planes of the sections ; the 
context is always sufficient to prevent ambiguity. 
