172 



MR. ROBERT FINLAY ON 



curve u is cut by the straight lines v and v' . For, since any 

 values of x and y which satisfy the simultaneous equations 

 (1) and (2) will also satisfy equation (4), it follows that the 

 curve Mj must pass through the two points in which the 

 curve u is cut by the straight line v ; and in a similar manner 

 it can be shown that the curve m, passes through the inter- 

 sections of u and v' . 



(a.) When the straight line v is without the curve w, the 

 points in which it intersects that curve will be imaginary; and 

 since the curve u^ passes through these points, it follows that 

 in this case v will be an ideal common secant of u and w,. 

 But when the straight line v cuts the curve u in two real 

 points, the curve m, will pass through these points, and v will 

 be a real common secant of u and m,. 



(6.) When a' = o and ^' = o, the straight line v' passes 

 to infinity, because the points in which it meets the axes of 

 X and y are at infinity. In this case we have «?'=!, and 

 equation (4) becomes 



u,^= u + kv ==: o (5). 



Hence we see that this equation represents a system of conic 

 sections having with the curve u a common secant at infinity. 

 When the curve u is an ellipse, this secant at infinity must 

 evidently be ideal. 



Since the terms containing the squares and product of x 

 and y are the same in equations (1) and (5), the curves u and 

 u^ are evidently similar and similarly placed. Hence any two 

 curves of the second degree which are similar and similarly 

 placed have a common secant at infinity. 



(c.) If a — a and /S' = i3, the straight lines v and v' will 

 coincide, and the four points of intersection of the curves 

 u and M, will coalesce into two points of contact. In this 

 case equation (4) becomes 



7/3 = 21 4- kv^ = o (6), 



which therefore represents a conic section having a double 

 contact with the curve u, the equation to the chord of contact 



