422 



47. It may be proved precisely as in Articles 7, 36, that the points 

 of contact of the pair of conies given hy the equation (85) with the 

 conies S-L*-0, S - M 2 = 0, $ - N=0, are constructed hy drawing 

 the three lines 



P(Z-M) _ Q(M-JSf) _ R{N-Z) 



X fi v 



"We give in the annexed scheme the entire system of forty-eight 

 lines for the sixteen pairs of equations : — 



1°. For the system of tangential conies (85)-(88) corresponding to 

 the system of concurrent common chords 



Z-M=0 M-N=0, iV-Z = 0, 

 the equations of the lines for constructing the points of contact are 



(101) 

 (102 

 (103 

 (104) 



P(L- 



X 



M) 



Q(M- 



■#) 



R{N- 



V 



-L) 



P(Z- 



X 



M) 



Q(M- 



■JV) 



R{N- 



L) 



P(Z- 

 X' 



M) 



Q(M- 



JY) 



R(N- 



L) 



P(L- 



M) 



Q(M- 



N) 



R{W- 



L) 



2°. For the system of conies (89)-(92) corresponding to the system 

 of concurrent common chords L + M= 0, M-N=0, N + Z = 0, the 



(105) 

 (106) 

 (107) 



P(Z + M) 



Q {M- 



N) 



R(N+Z) 



\ 









P(Z+M)_ 







R(isr+z) 



\ 







v\ 



P(Z + M) 



Q(M- 



N) 



R{N+Z) 



X\ 









P(Z + M) 





N) 



R(isr+z) 



X\ 









(108) 



3°. For the system of conies (93)-(96) corresponding to the system 

 of concurrent common chords Z + M- 0, M + N= 0, N- £ = 0, of the 

 conies £-£ 2 = 0, S- if 2 = 0, S- N 2 = 0, the equations of the lines 

 through the points of contact are, 



