68 
ANALYTICAL RESEARCHES CONNECTED WITH 
[114 
It may be remarked that these equations are satisfied by 
/8 = 0, /3' = 0, /3" = 0, y = 8, y = 8', y" — 8", 
or if we write 
a 7 a' a" 
V) > 77 
7 7 7 
the equations become by a simple reduction, 
F 2 + X 2 +2Z YZ = P -1, 
Z 2 + X 2 + 2 mZX = m 2 — 1, 
X 2 + F 2 + 2n XY = n 2 — 1, 
which are equivalent to the equations discussed in my paper “ On a system of Equations 
connected with Malfatti’s Problem and on another Algebraical System,” Cambridge and 
Dublin Mathematical Journal, t. iv. [1849] pp. 270—275, [79]; 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. 
Writing the second and third equations under the form 
A' + B'X + C' VT+x*=o, 
A" + B'X + C" Vl+X 2 = 0, 
the result of the elimination may be presented in the form 
A'A" + B’B" - C'C" = s/A' 2 + B' 2 - C' 2 VA" 2 + B" 2 - C" 2 , 
which is most easily obtained by writing X = tan (f> and operating with the symbol 
cos -1 ; but if the rationalized equations be represented by 
X + 2/jl'X + v’X 2 = 0 and + 2g"X + t/'X 2 = 0, 
the form 
4 (\V — fx 2 ) (\"v" — fx" 2 ) = (Vi/' + W — 2/x'/x") 2 
leads easily to the result in question. The values which enter are 
A' = a' +/3'Z, A" = a" +/3"F, 
B'=/3'+y'Z, B" =(3'' + y"Y, 
C = 8' *JT+Z 2 , C" = 8" Vl + F 2 ; 
whence, in the first place, by the equation connecting F, Z, 
ffC" = -^{a + p(Y+Z)+SYZ}.