[ 253 ] 
XIV. Analytical Researches connected with Steiner’s Extension of Malfatti’s Pro- 
hlem. By Arthur Cayley, M.A., Fellow of Trinity College^ Cambridge. Com- 
municated by J. J. Sylvester, Esq., F.R.S. 
Received April 12, — Read May 27, 1852. 
The problem, in a triangle to describe three circles each of them touching the two 
others and also two sides of the triangle, has been termed after the Italian geometer 
by whom it was proposed and solved, M.\lfatti’s problem. The problem which I 
refer to as Steiner’s extension of Malfatti’s problem is as follows : — ‘‘To determine 
three sections of a surface of the second order, each of them touching the two others, 
and also two of three given sections of the surface of the second order,” a problem 
proposed in Steiner’s memoir, ‘Einige geometrische Betrachtungen,’ Crelle, t. i. 
The geometrical construction of the problem in question is readily deduced from 
that given in the memoir just mentioned for a somewhat less general problem, viz. 
that in which the surface of the second order is replaced by a sphere ; it is for the 
sake of the analytical developments to which the problem gives rise, that I propose 
to resume here the discussion of the problem. The following is an analysis of the 
present memoir : — 
§ 1. Contains a lemma which appears to me to constitute the foundation of the 
analytical theory of the sections of a surface of the second order. 
^ 2. Contains a statement of the geometrical construction of Steiner’s extension 
of Malfatti’s problem. 
§ 3. Is a verification, founded on a particular choice of coordinates, of the con- 
struction in question. 
§ 4. In this section, referring the surface of the second order to absolutely general 
coordinates, and after an incidental solution of the problem to determine a section 
touching three given sections, I obtain the equations for the solution of Steiner’s 
extension of Malfatti’s problem. 
^ 5. Contains a separate discussion of a system of equations, including as a parti- 
cular case the equations obtained in the preceding section. 
§§ 6 and 7 - Contain the application of the formulee for the general system to the 
equations in ^ 4, and the development and completion of the solution. 
§ 8. Is an extension of some preceding formulae to quadratic functions of any 
number of variables. 
