SUPPLEMENTABY CURVES. 



173 



being v = o. When the straight line v is without the curve 

 u, the two points of contact are imaginary, and the chord of 

 contact is ideal. 



(d.) When a = o and ^ — o, the straight line v passes to 

 infinity, and equation (6) becomes 



u^= u -^ k = o (7). 



Now since this differs from equation (I) only in its absolute 

 term, it is evident that the curves u and u^ are concentric, 

 similar, and similarly placed ; and thus we see that any two 

 curves of the second degree which are concentric, similar, 

 and similarly placed, have a double contact at infinity. From 

 this it follows that any two concentric circles have a double 

 contact at infinity. In this case, as well as that of two similar 

 and concentric ellipses, the points of contact must evidently 

 be imaginary, and the chord of contact ideal. 



(e.) When the quadratic function u can be resolved into two 

 factors v^ and v^ of the first degree, equation (6) becomes 



W, z=: v^v.^-\- k v"" = o (8). 



In this case the conic section u breaks up into two straight 

 lines v^ and v^, and the curve u^ touches these lines at the 

 points in which they are cut by the straight line 'P. Hence 

 it is evident that the point of intersection of v^ and v^ is the 

 pole of the straight line v in relation to the curve u^. Now 

 when the lines v^ and «, are imaginary, their point of inter- 

 section is real, and continues to be the pole of the straight 

 line v; but the points in which v^ and v^ touch the curve u^ 

 are imaginary, and the chord of contact v is ideal. 



III. 



The theorems given in the last number relative to the 

 conic sections may readily be generalized. Thus, if m = o 

 be a curve of the nth degree, and if ?; = o and v' = o be two. 

 curves of the mth degree, where m is not greater than ^ n, 

 then u + k vv'= o will represent a system of curves of the 



