PROCEEDINGS OF SECTION A. 303 



From this we may learn the condition of tangency of Sj and 2, viz., 



3/ 3/^ 3/ 



and therefore /^ + /Ji + /^ = 0, 



V «o V 3o V 7o 



i.e., if ai/3iyi lie on the cubic — 



V/ «o ^ V' ^0 ^ V 70 ' 



the conies 2 and S, will touch, and the co-ordinates of the point of con- 

 tact will be (V^^, ^/jQiF. v^y^O- 



§ 4. On the curve ivhose equation is — 



The above curve may be easily shown to be the locus of points 

 whose axes of homology touch the conic S. 



A point on the curve may be expressed in any of the following 

 forms : — 



Va^ is} irj ■ l{ix - vf {v - \f (A - /.yd 

 ' Lav - ^f A^ - A)^ v\). - /x)d ' ' Le.- e^^ e^A ' 



The tangent at the point Ri (.oi -^.i - ^ ) is — 

 ^A' + Lx^ + U^ = 0, 



«o &0 7o 



which may be written — 



or e.(X.A,) + A/x,.Lo= 0. 



Hence as (T.Ai) has been found, we can consti-uct the point R, ; 

 the line joining R, to the intersection of (X.A,) and L^ will be the tan- 

 gent at Rj. 



The line joining Ri and Rj — the axis of which latter point is (T.Aj) 

 — is — 



^^A3(/. - .)3 + |/.3(, ._ A)3 4- 1 .XA - l^r = 0, 

 but this is the equation to the tangent to the above quartic at the point 



Hence the quartic may be regarded as the envelope of the line 

 joining the two points whose axes are the tangents to S at the extremities 

 of chords of S drawn through O. 



The lines (T.A3) (T.A4) are the axes of two points — 



^3 |_A-^(/x-.r t^{u-xy- .^A-z^rd ' ^^** Le?' e?' e?] 



on the quartic. The equation of R3 R^ i.s — 



} A3(/x - uf e,3 4- I ,.\u - Xf 0/ + J .^(A - ,.f 03^ = 0, 

 showing that R3 'R^ touches the quartic at the point — 



LAV- ^yei" i^\^ - ^fer ax - t,r e-\ 



