262 
MR. A. CAYLEY’S ANALYTICAL RESEARCHES CONNECTED WITH 
Or finally, the condition in order that the sections 
xs/ C+(Ax+%4-/2)Y4-zx/9[+>/ 
xs/ B +3/n/ (gr +^+ C2)Z + ^—cp^l+Z^w = 0 
(the former of which is a section touching z=0, x=0, and the latter a section 
touching x=0, y=0) may touch, is 
/YZ+^/S(Y+Z) + (/+2 Vi) -yi^-/iTP\/T+Z’=o. 
The preceding researches show that the solution of Steiner’s extension of Mal- 
FATTi’s problem depends on a system of equations, such as the system mentioned 
at the commencement of the following section. 
^5. 
Consider the system of equations 
a+/3(Y+Z)+yYZ4-^N/T + YVi+^'=d 
a'-j-|3'(Z + X) + y'ZX+^'x/lT^\/T+X^=0 
a"+j3"(X+ Y) +/XY4 -^Vi+XVT+Y^=0. 
These equations may, it will be seen, be solved by quadratics only, when the coeffi- 
cients satisfy the relations 
13 /3" 
|82 + y _ §2 jS'2 ryH _ gf2 _ gf/2 
2 2 
y 
."2_„//2 ’ 
equations which it should be remarked are satisfied by 
(3 = 0, (B'—O, |8"=0, 7=^, y'=^', y"=^". 
Or if we write 
« a! a!' 
ry ^ y 
the equations become by a simple reduction, 
Y^+Z'^+2/YZ=/^-l 
Z^ + X^ -1- 2mZX = ?7i^ — 1 
X^-fY^^-2y^XY=w^-l, 
which are equivalent to the equations discussed in my paper “ On a System of Equa- 
tions connected with Malfatti’s Problem and on another Algebraical System,” Cam- 
bridge and Dublin MathematicalJournal, t. iv. p. 270; the solution might have been 
effected by the direct method, which I shall here adopt, of eliminating any one of the 
variables between the two equations into which it enters, and combining the result 
with the third equation. 
