420 



Let these chords (fig. 8) intersect in 0; then from 0 draw pairs of 

 tangents to the conies ; then it may be proved — but I shall not occupy 

 space in doing so — that the six points of contact, a, a' ; b, b' ; c, c' , are 

 in the circumference of a conic, and denoting two of the anharmonic 

 ratios, as in Art. 40, of the points 



a, a, b, b' } by X, X' ; 



b, b', c, c', „ fi, fi ; 



c, c', a, d, „ v, v ; 



and denoting the results of substituting the co-ordinates of the poles of 

 the chords of contact L, M, N, in the equations of the conies S - L 2 = 0, 

 S- M 2 = 0, S- N 2 = 0, by P, Q, R, respectively, we have from equa- 

 tion 84 the following system of equations of pairs of conies, each conic 

 having double contact with S, and touching 8 - L l = 0, S- if 2 - 0, 

 £_i\r 2 = 0:— 



|X(#- 





p 





L) 



p 





L) 



p 



X'(£*- 



-L) 



(85) 



q y r 



9 . (86) 



' y q • y it 



(87) 



(88) 



43. The system of circles a', ft, 7, have corresponding to them the 

 system of conies Si + L = 0, S* - M = 0, S* - JV= 0 ; and denoting by 

 Xi, X'j, /ij, v t} v\, quantities analogous to X, \ f , /*, p! t v, %>', of the 

 last article. 



The common chords are L + M- 0, M- N=0, N+ L = 0 ; and the 

 system of equations is 



j wtt) • jm^m) + __ 0; (89) 



^tx-) + + JM*EZ> _ 0 ; (90) 

 j^±| . j^ St k^_ 0| (91) 



