419 



where I, V, are the squares of the sines of half the direct and half the 

 transverse common tangents of S r , S". 



40. If the circles a, /3, 7, be cut orthogonally by J, and denoting 

 two of the anharmonic ratios (Art. 39) in which 



J iutersects /3, 7 by \, V, 



we have 



I - X tan r' tan r", 

 I - X tan r ' tan r" ; 



and substituting this value of I and corresponding values for m and n in 

 equation (70), it becomes transformed into 



igsER pgnrg) /3CT) = 0 . (84) 



\ sinr + \ sin/ + \ sinr" ' K } 



and the equations of the other pairs of circles are got from this by pro- 

 perly accenting X, fi, v. 



41. In equation (84) it will be observed that \ fi y v, are anharmonic 

 ratios, and that sin r, sin sin r", are the results of substituting the 

 co-ordinates of the poles of the great circles L, M, iV, in the equations 



£-Z 2 = 0; £-JP = 0; £-iV 2 = 0. 



These considerations will enable us to write down the equations of the 

 conies in pairs having double contact with the conic S, and touching 

 the three conies 



S-L* = Q; S-M* = 0; S-W^O. 



42. Since the equations S-Z 2 =0, S - M 2 = 0, S-N 2 = 0, are 

 the same analytically as the equations of conies having double contact 

 with a given conic, we can, by means of Arts. 40, 41, write down the 

 equations of the conies in pairs having double contact with a given 

 conic, and touching three others also having double contact with the 

 same given conic. Thus, corresponding to the system of circles a, /3, 7, 

 we have the conies 



£*-Z = 0, ^ - M= 0, £*--ZV=0, 

 whose common chords are (see Salmon, page 228) 



Z-M=0, Jf-i\T=0, W-L = 0. 



