334 



N ^ M 



F — B C — — . 



M—N M—N 



In like manner the points A, and A\, J, and .4', are conjugate 

 points of S, therefore we have also 



L N 



G=C 

 H = A 



N-L 

 M 



A 

 B 



N-L 



L 



L-M L — M 



and the equation of the system itself may be written in the form 



A%. 



2M 

 L-M 



2N ' 



N—L 



+ ^5, 



+ ^§. 



2L 

 L-M 



2L 



§x-^, 



27V 

 M—N 



§. 



+ 





Since S contains the three degenerate conies ^iV'i, IsV'j» sb^j' i^ 

 is seen that the expressions between brackets in the above equation 

 denote the right lines ipj, tp,, V'b» and we may write 





" 2L 

 L^I 



2M 



2N 



-M^' ^ N- 



2N 



f _f 



§. 



M—N 



(2) 



2L 



2M 



N-L^''^ M—N^ 



,-5, I 



where P^, P^, P^ are determinate constants. 



From these equations we deduce, always using the coordinates 

 si> >,,§,, the coordinates of the points B^, B,, Bf, the centres of the 

 conies ^,\p„ 2,ip„ §,tf>,. 



Putting 



L{M-N) = q^ , M{N—L) = q, , iV (L-J/) = 9,, . . (3) 



so that the constants c[^, q,, g, are related by the equation 



, 9i + 9, + ^i = 0, 

 we find for the coordinates of B^, B^, B^ respectively (0, q^, q^), 

 (5'i. Oj ?i)> (?». 9i. 0)- Incidentally we may remark that at the points 

 jBj, B^, Bt the right lines i^, §,. |, are touched by the conic 



and at the same time we conclude that the equation of the conic //, 

 that forms part of the Jacobian, must be 



For plainly this conic passes through the points A^, A,, A, and 

 also through B^, B„ B^. 



The equations of the tangent '§^ and of the right line \p^ remain 



