MATHEMATICAL SECTION. 125 
and the equation of a normal plane at the surface point (2, y, z) 
and passing through (2,, 7, 2,), (not necessarily a surface point, but 
considered so here), is: 
0=[ ¢—»(%)-¢-9(&) |Lo-»(Z)-@-9() ] 
—[o-»(z)-«-9(%) |Le—9(@)-«-9(@) ] 
=[%—y) €—2)-@—2G—-y)] (=) 
+[@—)G—-) —%—» €-a1() 
du 
+ [@—2) €-2) -@-DE-a1(F) © 
If in this we replace the surface point (2, y, z) by the surface 
point (2,, y,, z,) and (, 7, ¢) by the surface point (a, y, z) we obtain: 
du 
= (G.— 9) @— 2) —@—«) y-wi(F) 
du 
+ia— 4) — %)—-G—n) @—4)] aa) 
“-FLG@,— %,) G—2)— (Ce 2) (x 7 2,)] (37) (4) 
which, if combined with the equation of the surface, gives the nor- 
mal section at (2, y,, 2,) through (2,, y,, 2). 
If, however, we replace in (3) (&, y, £) by the surface point 
(a, y, 2) we obtain: 
0=[(y— y) @{— 2~)—(@, — 2) (y—y)] 3) 
+{[@-9M-y) -—G-NG-4] (iz) 
+ —2) @—2)-@-)@—29) (7) 6) 
and this, combined with the equation of the surface, gives the dior- 
thodic curve. 
As we move along the diorthode, (5) may be considered a plane 
which turns about the chord (1, 2) as an axis, so as to be always 
normal to the surface. It follows that the normals at any point of 
the diorthede are constrained to pass through the chord. They will 
thus generate a ruled surface, whose equation is not (5) however. 
