46 
ON schellbach’s solution of malfatti’s problem. 
[170 
and consequently 
where if 
v 2 +£ 2 = ^s 2 -2\l, v £=s(\-l), 
cj) = —-, (X — <j>) 2 = ^s 2 + (¡r — m 2 — n 2 , 
i.e. 
2 mn 
s 
4 m 2 n 2 
40 -1 m 2 — n- 
^ /{/ (s 2 — 4m 2 ) (s 2 — 4/i 2 ) 
Hence 
. , 4tlmn l /-—-— /——-— 
17“ + = y s 2 — vs 2 — 4m 2 vs 2 — 4?i 2 , 
??£■ = 2wmi — sl + Vs 2 — 4 m 2 Vs 2 — 4A 2 , 
g* + f = is 2 - - - Vs 2 - 4n 2 Vs 2 - 4/ 2 , 
s s 
££ = 2nl — sm + ^Vs 2 — 4?i 2 Vs 2 — 41 2 , 
p + = i.s 2 - - - Vs 2 - 4i 2 Vs 2 - 4m 2 , 
s s 
%t) = 2Im — sn + \ Vs 2 — 4/ 2 Vs 2 — 4m 2 , 
which is in fact at once deducible from the formulae in my paper “ On a System of 
Equations connected with Malfatti’s Problem and on another Algebraical System,” 
(Carnb. and Dvbl. Math. Journ. t. iv. (1849), p. 270 [79]). 
Write now for l, m, n, rj, their values in terms of a, b, c, x, y, z. We have 
(£« - y)~ + (1« - z? - 7 (i* - a) (1« - y) (¿S -z)= Is 2 - (f s - a) 2 , 
i.e. 
y 2 + z 2 Qs — a) yz — 2a {y + z) + a 2 = 0, 
or reducing 
and we have thus the system 
(y + z - a) 2 - 4 (l - ^ yz = 0, 
y + z + 2 \ - ^ Vyz = a, 
z + x + 2 u --^Vzx =b, 
x + y + 2 Vxy = c,