60 
ANALYTICAL RESEARCHES CONNECTED WITH 
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the values of a, 6, c remaining to be determined. JSIow before fixing the values of 
these quantities, we may find three sections each of thexli touching a determinator at 
a point of intersection with the section which corresponds to it of the sections 
cy — bz = 0, az — cx = 0, bx—ay = 0, and touching the other two of the last-mentioned 
sections; and when a, b, c have their proper values the sections so found are the 
tactors. For, let Xx + py + vz + pw = 0 be the equation of a section touching the deter 
minator — ax + ~ y + ^ z + w = 0, and the two sections bx — ay — 0, az — cx — 0: and 
suppose 
A 2 = X 2 + /a 2 -f v 2 — 2pv — 2vX — 2Xp — 2p 2 ; 
the conditions of contact with the sections bx — ay = 0, az — cx — 0 are found to be 
(6 + a) A = (6 + a) X — (b + a) p — (6 — a) v, 
(c + a) A = (c + a) X — (c — a) y — (c + a) v, 
values, however, which suppose a correspondence in the signs of the radicals. Thence 
(b + a) fji = (c + a) v; or since the ratios only of the quantities X, y, v, p are material, 
p, = c + a, i> = b + a, and therefore 
A 2 — X 2 — 2 (2a + b + c)X + (b — c) 2 — 2p 2 , = (A — b — c) 2 , 
or pi 2 — — 2 (aX + be). 
Hence the equation to a section touching bx — ay = 0, az — cx — 0 is 
Xx +(c + a) y + (b + a) z + V — 2 (aX + bc) \w = 0 ; 
and to express that this touches the determinator in question, we have 
± a (X — b — c) = X — a (2a + b + c) + 2^/— 2 (aX + be); 
and selecting the upper sign, 
- X — 2aa = — 2 V — 2 (aX + be); 
CL 
whence 
X = — 2a (aa — V — 26c), V — 2 (aX + be) = (2aa — V - 2be) ; 
or the section touching the determinator and the sections bx — ay = 0, az - cx = 0 is 
— 2a {aa — V — 2be ) x + (c + a) y + (6 + a) z + (2aa - V — 26c ) w = 0 ; 
and at the point of contact with the determinator 
-® + ^y + 2 V +w=0 ' 
2 yz + 2 zx + 2 xy + w 2 = 0.