148 
ON THE POKISM OF THE iN-AND-CIRCUMSCRIBED TRIANGLE, [129 
shows that if x 2 , y< 2 , z 2 are such that x. 2 2 + y. 2 2 + z,? = xi + yi 4- zi, then that also 
ax 2 2 + by 2 + cz<? — axi + byi 4- cz 2 . I proceed to determine © so that we may have 
x? + yi + zi = xi + yi + z x . We obtain immediately 
O2 2 + yi + z-i) = (lW + m*yi + »V) (<*V 4- fi 2 yi 4- 7V) 
— (a. 2 l 4 xi + /3 2 m 4 yi 4- 7 2 %%j 4 — 2/3ym 2 n?y 1 2 z 1 2 — 2yan 2 l 2 z 2 x{- — 2a(3l 2 m 2 xiyi); 
write for a moment 
a#! 2 4- ft^ 2 4- ezi =p, xi 4- yx 2 4- z 2 = q, so that a. =p — aq, /3 =p — bq, y=p — cq, 
then 
a 2 x x 2 4- (3 2 yi + rfz 2 = qp 2 — 2p . pq + {a 2 xi + b 2 yi + c 2 zi) q 2 , = q {{a 2 x 2 4- b 2 y 2 4- c 2 z 2 ) q — p 2 }, 
= q{(b — c) 2 y 2 z x - + (c — a) 2 zixi + (a — b) 2 x x 2 y(], 
a 2 l 4 xi 4- (3 2 m i y 1 4 ‘ 4- y 4 n 4 z 4 — 2/87 m 2 n 2 y 2 z 2 — 2ya.n 2 l 2 z 2 x 1 2 — 2a(3l 2 m 2 x l 2 y 1 2 
= p 2 \t 4 x x 4 4- m 4 y( 4- w 4 ^ 4 — 2m 2 n 2 y 2 z 2 — 2n 2 l 2 z 2 x 1 2 — 2l 2 m 2 x 1 2 y 1 2 } 
— 2pq [al 4 xi + bm 4 y( 4- cn 4 zi — (b + c) m 2 n 2 yizi - (c + a)n 2 l 2 z 2 xi — (a 4- b) l 2 m 2 xiyi\ 
4- q 2 [a 2 l 4 xi 4- b 2 m 4 yi + (?n 4 z x 4 — 2bcm 2 n 2 y 2 z 2 — 2can 2 l 2 z^x 2 — 2abl 2 m 2 xiyi\, 
the first line of which vanishes in virtue of the equation lx x + my 2 + nz 2 = 0; we have 
therefore 
( x 2 2 + yi + Z 2 ) -5- ( x i + yx 2 + z 2 ) 
= (l 4 xi + m 4 yi 4- n 4 Zx 2 ) {(6 — c) 2 y 2 z 2 + (c — a) 2 z 2 x? 4- (a — b) 2 xiyi] 
4- 2 (axx 2 4- by 2 4- cz x 2 ) [al 4 xi 4- bwiyi 4- cn 4 z(—(b + c) m 2 n 2 yizi— (c 4- a) n 2 l 2 Zx 2 Xx 2 — (a + b) l 2 m 2 xiyi) 
— ( x \ + yi + z O {p?l ix i + b 2 m 4 y 4 4- c 2 n 4 zi — 2bcm 2 n 2 y 2 Zx — 2can 2 Pz 2 x 2 - 2abl 2 m 2 x 1 2 y 1 2 }. 
Hence reducing the function on the right-hand side, and putting 
(#2 2 + y? + zi) -T■ (Xx 2 + yi + zi) = 1, 
we have 
= a 2 l l x a fi + b 2 m 4 yx + c 2 n 4 Zx 
+ (c 2 m 4 — 2b 2 m 2 n 2 ) yizi + (a 2 n 4 — 2c 2 n 2 l 2 ) zi x i + (bH 4 — 2 a 2 l 2 m 2 ) x x 4 yi 
+ (b 2 n 4 — 2c 2 m 2 n 2 ) yiz 4 + (cH 4 — 2a 2 n 2 l 2 )zixi + (a 2 m 4 — 2b 2 l 2 m 2 )xiyi 
+ {l 4 (b — c) 2 + m 4 (c — a) 2 + n 4 (a — b) 2 
+ 2m?n 2 (be — ca — ab) + 2n 2 l 2 (—be + ca — ab) + 2 l 2 m 2 ( — be — ca + ab)} ociyizi. 
The value of © might probably be expressed in a more simple form by means 
of the equations lx x + my x + nzx = 0 and l 2 bc + m 2 ca + n 2 ab = 0, even without solving 
these equations; but this I shall not at present inquire into.