79] 
465 
79. 
ON A SYSTEM OF EQUATIONS CONNECTED WITH MALFATTI’S 
PROBLEM, AND ON ANOTHER ALGEBRAICAL SYSTEM. 
[From the Cambridge and Dublin Mathematical Journal, vol. iv. (1849), pp. 270—275.] 
Consider the equations 
by 2 + cz 2 + 2fyz = 6 2 a (be — f 2 ), 
cz 2 + ax 2 + 2gzx = 6 2 b (ca — g 2 ), 
ax 2 + by 2 + 2hxy = 6 2 c (ab — h 2 ); 
or, as they may be more conveniently written, 
by 2 + cz 2 + 2fyz = 6 2 a%X, 
cz 2 + ax 2 + 2 gzx = 6‘ 2 bd3, 
ax 2 + by 2 + 2 hxy = # 2 c©. 
The second and third equations give 
(g 2 Q.D — A 2 23) x 2 — b 2 33y 2 + c 2 (&z 2 + 2cgO&zx — 2bli$$xy = 0, 
hence {(g 2 dD — /¿ 2 23) x — blj&y + cg(&z) 2 — 23© (— bgy + chz) 2 = 0, and consequently 
(g 2 © — /¿ 2 23) x — &V23 OV© + /i\/23) y + cV© OV© + W23) 2 = 0: 
dividing this by © + Aa/23, and writing down the system of equations to which the 
equation thus obtained belongs, 
(¿/V© — W23) x ~ 5\/33 y 4- c\/©^ = 0, 
— /V©) y — Ca/© 2 = 0, 
-flu/gt a?+ 6a/23 .y + (/V23 — ^a/^I) z = 0. 
c. 
5.9