7] LINES OE CURVATURE OF AN ELLIPSOID. 39 
From the equations (14), (29) it is easy to prove the well-known form 
X 2 y 2 z 2 _ 
a 2 + 0 + b- + 0 + We ~ 1 ( 30 ) 5 
in fact, multiplying (29) by m, and adding to (14), we have the equation (30), if the 
equations 
1 b 2 -c 2 \ 1 
a 2 + m B 2 -C 2 a 2 ~ aWe ’ ( 81 )> 
1 c 2 —a 2 1 1 
6 2 ^ 6' 2 - 4 2 6 2 b 2 + 0 ’ 
1 , a 2 — b 2 1 1 
— _i_ 
c 2 A 2 - c 2 (c 2 + 0) ’ 
are satisfied. 
But on reduction, these take the form 
{B 2 — C 2 ) 6 + (b 2 — c 2 ) m0 + ma? (b 2 — c 2 ) = 0, (32), 
(C 2 — A 2 ) 0 + (c 2 — a 2 ) 7ii6 + mb 2 (c 2 — a 2 ) = 0, 
{A 2 — B 1 ) 6 + (a 2 — b 2 ) md + me 2 (a 2 — b 2 ) = 0, 
and since, by adding, an identical equation is obtained, in and 0 may be determined 
to satisfy these equations. The values of 0, m are 
a _ (a 2 - b 2 ) (b 2 - c 2 ) (c 2 - a 2 ) 
a 2 (B 2 - C 2 ) + b 2 (G 2 - A 2 ) + c 2 (A 2 -B 2 ) 
b 2 c 2 (B 2 - (7 2 ) + c 2 a 2 (G 2 - A 2 ) + a 2 b 2 (A 2 - B 2 ) 
m (a 2 — b 2 ) (b 2 — c 2 ) (c 2 — a 2 ) 
(34).