64 



KANSAS UNlVF.RSriY yUAKTEKLY, 



A= 



IB 



B, 



|B« 



IB 



r^ 



B„ 



A 



kA, 



k,A. 



B 



kB^ 



k,B^ 



I 

 k 

 k. 



lA I 



:a, I 



Ag I 



lA I 

 A, I 



|a, I 



A I 



A, I 

 A„ I 



A 



kA, 

 kjA^ 



B 



kB, 

 k,B. 



k 



ABA 



|A, B, kA, 

 lAo B^ k^Bg 



ABB 

 A, B, kB, 

 A, B, k^Bo 



:A B : 

 A,B, k 

 A,B,k, 



ABC 

 A,B,C, 



A., B., C. 



Expanding A, B, etc., wc have 



A=:k,A,(B— BJ j-kA,(B,— B) + A(B,~B2), 



A,r=.k,B„(B— BJ + kB(B, B) + B(B,— Bg), 

 A3=k,(B— B,Vfk(B2— B) + (Bj— B„), 



B=k,A,(A— AJ + kA,(A3-A) + A(A,-A,), 

 B,-:k,B,(A--A,)+kB,(A3— A) + B(A,— A„), 

 B,=.k,(A— A,)4-k(A2-A) + (A,— A,), 

 C^kjAgCAB,— A,B) 1 kA^CAjB— AB2) + A(A,B,— A2B,), 

 Cj=.k,B2(AB,— A,B) fkB,(AgB— AB,)+B(A,B2^A„B,\ 

 C. .:k,(ABi— A,B)-|-k(A2B— AB.,) + (A,B,— AgB,). 



k,A., kA, 

 kjBj kB, 



BlxlA 

 I 



B-B 



A- 

 |AB, 



=.kk 



B„— B B,— B, 



Aj A*— A A,— A.^ 



A,B A„B— AB., A,B„— A.B,! 

 A B 13 



[A, A, I . 

 !A., B., I 



3. Obtain the explicit normal form of t3pe II by solving the 

 equations of the implicit normal form for x, and y,. 



4. Show that the determinant of tht 

 type II is given by 



explicit normal form of 



5. Obtain the explicit normal form of type III by solving the 

 equations of Art. 17 for x, and y, and show that the determinant 

 of the explicit form is ^^=1. 



6. Show that T" leaves invariant besides the conic K the pencil 

 of conies to which K belongs, all having contact of the third order 

 at A. 



