114] 
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114. 
ANALYTICAL RESEARCHES CONNECTED WITH STEINER’S 
EXTENSION OF MALFATTI’S PROBLEM. 
[From the Philosophical Transactions of the Royal Society of London, vol. cxlii. for the 
year 1852, pp. 253—278: 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, Malfatti’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,” Grelle, t. i. [1826 
pp. 161—184]. 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. 
C. II. 
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