308 Transact io/m. 



Comparing this witli tlie form of S given above we have 



hence the equations of the four chords of contact with Lj and 

 La of the conies which are the inverses of the connxion 

 tangents of Si and S.^ are 



The inverses of the points in which these four hnes meet 

 Li and L.2 are the points of contact of the common tangents 

 of Si and S2. 



Let Ci = \/Zi/2a+ v'?;ii'»?2/3 + \/7ljfl^y = 

 C^ = \/lil.2 a — v^Vli^m^ /8 — v/^ijWa y = 



C3 = — •yiil^a + "^m^in^fi — s/n-^n.^y = 

 C4 = — ^yiJi a — ^'iiiim^ )8 -f- yn-^iio, y = o 

 and form the conic 



Ti = L,L, - Ci'^ = 

 which is the inverse of a common tangent t^. 

 Now write 



Pj = K^mjn^— ^/m^Ui P2 = N/mi??2+ V'woi/i 

 Qi ^ VnJ^ - VvJ^ Q, = VnJ.2 + ^'^1 



El = \/limi — y/Un^ K., = vliin^ + v'^vii 



Then the conic Ti reduces to 



On inversion we obtain tlie four common tangents of Si and S2 



^i=PA + Qi^^ + Ei'^y=0 

 ^, = P2^a + Q.//5 + Ei'V=0 



To find the co-ordinates of the points of contact of ti with 

 Si and S.j, solve for a/3y between c^ and Li and Cj and L, and 

 invert. 



We thus find that ^ will touch Sj and S.^ respectively in the 

 points 



VTT "or "Ri^ V Pi Qi Ri 



with similar expressions for the points of contact of t.,, ^i, and t^ 

 with these conies. 



