MATHEMATICAL SECTION. 127 
The following diagram will illustrate the relative course of these 
curves: 
“normal section 
tteiete dioTthode 
Bee proorthode 
te brachisthode 
Any surface of the second degree may be represented by 
z—a\? yp 2 
= 0 = |—— v—_|  — 8 
u=o= (**) 404% a (8) 
The origin is taken at one of its real vertices, so that (a, 0, 0) is 
its centre. The equation of the diorthode is then by (5), if we 
write %,— ©, = AX; J, — J, = AY; 4-444 
0=[y— 9) (4-2) —@—2)(U—- DIG 
C—O 
+[e%—2)m%—-Y)—-Y—-YNa-AI | 
+[@%—2)4-)-&@-%) GA- «“)] = 
Zz 
= (y, X, — YL, + YX — Ay) ”q 
“—a 
+ & % — 4% + 24y — ydz) — 
+ (a, 2 — 2, % + waz — zhx) = 
1 
ns ale 2)n sa (2 1) 
eta 
+ (2, % — % Yo) a 
2 Z 
+ (4, % — 2, 2, + pdz) z + (YX = Yy % — VIG (9) 
The equations of the chord (1, 2) may be written: 
brat RL Se Pea AR (10) 
Every point of the chord, therefore, satisfies (9), and since that 
