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XXXI. On the Problem of the In-nnd-circnmscribed Triangle. 

 By the Rev. George Salmon, Trinity College, Dublin*. 



THE following was suggested by Mr. Cayley's paper in the 

 January Number of the Philosophical Magazine, which I 

 have just met with. 



Let it be required to find the envelope of the third side of a 

 triangle inscribed in a conic U, and two of whose sides touch a 

 conic V. 



Let the condition that XU + V should represent two right 



lines be 



vx»+©x2+e'x+v'=0; 



then since the value of X plainly cannot depend on the particular 

 axes to which the equations are referred, it follows, that, no mat- 

 ter how the equations are transformed, the ratios of the coeffi- 

 cients of the powers of X in the equation just written remain 

 unaltered. 



Let now the sides of the triangle in any position be x, y, z, 

 then the equations of the conies admit of being transformed into 



U = 2xy+2yz + 2xz=0, 



V = /3^2 ^ m^y"^ + w V — ^Imxy — 2mnys — 2nlsx—2Axy = ; 

 and it is plain that the equation 



AU+V=0 



represents a conic touched by the third side z. 

 But in this case we find, if l+m + n=p, lmn=r, 



V=2, = -/-2A, Q'z=z2p{2r+An), v'=-{2r+An)% 



whence 



40v'-0"=8A(2r + Aw)2; 



and the equation AU + V = may be written 



(4© V'- O'nU-4VV'V=0. 

 The coefficients in this equation being invariants, it follows 

 that the conic which we have proved is touched by the third side 

 is a, fixed conic. 



We can in like manner find the locus of the vertex of a tri- 

 angle circumscribed about V, and two of whose vertices move 

 along U. In this case the equations may be transformed into 

 U = 2xy + 2yz + 2zx + As\ 

 V =:: Px'^ + m^y'^ + n^z^ — 2lmxy — 2mnyz — 2nlzx, 

 and we have 



V = 2-A, = — / + 2/mA, Q'=4pr, v'=-4r2. 

 * Communicated by the Author. 



