174 



MR. ROBKBT FINLAY ON 



7ith degree, passing through the points in whicli the curves 

 V and v' intersect the curve u. Again, when the curves v and 

 v' coincide, the 2 mn points of intersection unite into mn 

 points of contact; so that the equation u + A y* = o repre- 

 sents a curve of the nth degree touching the curve u at mn 

 points which lie on the curve of contact v. It is evident also 

 that all this reasoning holds good when any of the curves 

 u, V, v' breaks up into a system of straight lines. 



IV. 



Let u-Aj/'-\-2Bx7j-\-Cx' + 2Dy+2Ex+Y=:o (1), 



and u'=Ay+2B'xy-^Cx''+2'D'i/ + 2E'x-i-W= o (2), 



represent any two conic sections; then, if k denote a given 

 constant, the equation 



u^ = ii -{■ k u'= o (3) 



will evidently represent a conic section passing through the 

 four points of intersection of the curves u and u\ Putting 

 A + ^A'=:A,, B + k B'==B,, &c., equation (3) maybe 

 written in the form 



A,f/' + 2B^x2/ + C^x' + 2'D^y + 2E,x + F^^o (4). 



Now if k be so assumed that the first member of this equa- 

 tion can be resolved into two factors of the first degree, the 

 curve Mj will break up into two straight lines, which will be 

 conjugate common secants of the curves (1) and (2). In this 

 case, let x^ and y, be the co-ordinates of the point of inter- 

 section of the two lines in question ; then, by transferring the 

 origin of co-ordinates to this point, equation (4) becomes 



A,y' + 2B,xy + C,x^+2'D,y + 2E.^x+l\ = o (5); 



where, for the sake of brevity, we assume 



E.= B,y, + C.a.,+ Ej ^^^' 



^.^Kl/:+^B,x,y,+ C^x:+2'D^y^+2E,x, + E, (7). 



"Because the two straight lines represented by equation (5) 

 pass through the new origin, we must evidently have 

 D, = o, E,=:o, F =o; 



