NORMAL FORMS. 103 
the perpendicular distances from P, and P to the sides of the 
invariant triangle. They express the fact that the cross- 
ratios of these perpendiculars are constant for all pairs of cor- 
responding points. 
130. Explicit Normal Form of Type I. Equations (10) are 
linear in x, and y,, and may be solved for these quantities, 
giving us the explicit normal form of T. To solve these 
equations we proceed as follows: Let the equations of the im- 
plicit form be written 
amy Gye al ae = all 
ANOBLING AN BA 
Alp Plertldnl yen, Aen, iy 
ci yl ‘| Dd? abit AH D 
A’ Br 1| A! Bt 
A” BY 1\ A” Bl 1 
Expanding and collecting ; 
w, | D(B-B") ~kN(B - BY) | _y, | D(A-A")- kN (A'— A") | 
= — D(AB!— AB) + kN (A/B!— A"B’), 
a, ) D(B- B’) WN (B’-B") { — y,} D(A- A) -wN(A'— AY | 
= — D(AB!— A/B) +N (A'B" — A"B) . 
Solving by determinants ; 
|e wy if @ 
[eAle SBiee ot | WALES ap eilA 
D\A' B 1\(AD—A'KN+A'KN’) AU By i KAR 
; | A” BY 4 ar Be 7 WA"! 
i AP SBM 1 le 7] 2 Y 
p\|A4’ B 1 (D—kN + k!N’) AgusB a pies 7 
| All B" 1 | A’ B i k 
Al BY 1 ke 
(11) 
1 y 1 0 | 
A B 1! A YBa 2128 
D| A’ B 1|(BD-BkN+B’KN’) WAL tReet IB! 
(ABU AC BUSS tee Bi 
Yi |A Ba - |e @ ff @ 7 
D|A' B 1 (D—kN+K'N’) WAL) SOR miata 
A” Bl 7 Al Be Ti Kk 
Av Bla i 
