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system of principal chords will be determined. But we shall 
obtain a more symmetrical result by assuming for the un 
known quantity 
c + b'm -f- an = s, or s — c = b'm + an, 
then am + cn + b' = ms, or m (s — a) = cn + b' 
bn -f cm + a' = ns, or n (s — b) = cm + a 
Hence, determining m and n from the two latter equations, 
m {(s - a) (s - b) - c 2 { = b'(s — b) + a c I 
n {(s — h) {s — d) — c' 2 | = a {s — d) + b' c'i 
And substituting for m and n in the former, we have 
(s — a) (5 — b) (s — c) — d 2 (s — a) — b' 2 (s — b) — c :i (s — c) — Za'b'c = 0, 
or s 3 - (a + b + c)s 2 + (ab + ac + - d~ — b'~ — c 2 )s 
— (abc — aa l — bb'~ — cc >2 + 2a'b'c) = 0. 
This equation, being of an odd degree, will always have 
one real root which substituted in (2) will give real values 
for m and n; and therefore in every surface of the second 
order there is at least one principal plane, or, which is the 
same thing, one system of principal chords. Also there can 
not be more than three, unless the particular form of the 
proposed equation of the second order should render any two 
of the equations (l) identical, in which case m and n would 
be indeterminate, and the number of principal planes would 
be infinite. 
Con. That the cubic has all its roots real, may be shewn 
by putting it under the form 
(s - c) | (.s - a) (s - b) - c 2 1 - | d~(s - a) + h' 2 (s - b) + 2 a'b'c' ^ = 0, 
and substituting for s, a and ¡3 the roots of (s-a)(s—b)-c 2 =0. 
The results of these substitutions, since (« — a) (a - b) = c' 2 , 
(a - ¡3) (b - /3) = c 2 , are 
— {o’\/a - a ± h's/a - b\ 3 and -f {a'\/d - (3 i b'\/.b - (3\ 2 ,