NEWSON: I'KOJECriVK TRANSFORMATIONS. 



EXERCISES. 



63 



I. Obtain the explicit normal form of a transformation of type 

 I from the implicit normal form by solving equation (4) for Xj and 



I^et the equations of the implicit form be written 



lA B 

 IA2 B, 



Ia. b.. 



\A B I 



N iA, B, i|_ N^ 



|A,B, I. 

 lA, B„ 1 



(1) 



Expanding and collecting: 



^^rJl^CB B,)^kN(B,^B,)[-y, -j D(A--A,)-k,N,(A,-~-A,) ^ 



=— D(AB,— A3B) + kN(A,B2— A.Bj) 



X, ] D(B-B0-1mN,(B -B, [ ^-y, \ D(A-A,)k,N, (A, ^A,) }- 



= ~D(AB,™A,B) I k^N,(A,B,-A,BO. 



jA B 

 DjA^B, 

 1A„B„ 



A B 

 A, B 

 A,B., 



I) A, Bj 



X y I o 

 A B I A 



|(AD-A,kN + A,k,N,) A^B, I kA, 



A,B, I k.A. 



(D— kN + k,N,) 



(ii) 



y-. 



X y I o 



A B I B 



A, B^ I kB^ 



A„ B, I k,B, 



X y I o 



• A B I I 



[Aj B, I k 



!A,B, I k, 



(iii) 



2. Show that the determinant of the explicit normal form of T is 



lA B I 3 

 A=kk, A, B, ij. 



|a„b, i' 



