519] ON CURVATURE AND ORTHOGONAL SURFACES. 3] 5 
which, on account of (A,. .$X, Y, Z)- = 0, and A + B + G = 0, reduces itself to 
(A,..\a,...).V\ 
49. We have 
Aa + Hh -f Gg = ä (2hZ — 2gY) 
+ h( gX- fY — (a — b) Z) 
+ g(fZ- hX-(c-a)Y) 
= X (gh - hg) 
+ Y (ag — gä — (gä +fh + eg)) 
+ Z (hä — ah + (hä + bh+ fg)) > 
or, observing that in the coefficients of Y and Z the second terms each vanish, this is 
Aä + Hh + Gg = X (hg — gh) + Y (ga — äg) + Z (äh — ha) ; 
and similarly 
Hh+Bb + Ff= X (bf -fb) + Y(fh — hf) + Z (hb - bh), 
Gg +Hf+Gc = X (fc — cf) + Y (cg — gc)+ Z (gf-fg). 
Adding these equations, the coefficient of X is the difference of two expressions each 
of which vanishes; and the like as regards the coefficients of Y and Z\ that is, we have 
and consequently 
2 
8X, BY, BZ 
X, Y, Z 
BX, BY, IZ 
(A,..Ja,..) = 0; 
= ((4),..$5,...) = -M,...pV, S7, SZ)\ 
the required relation. 
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