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6. We shall now examine the geometrical interpretation of the equations (8), first, for 
the sake of illustration, in special cases, and then we shall give the general results. 
We may remark in passing that all the contravariants of curves can be expressed in 
terms of these tangential curves ; for instance, if U be a cubic, the envelope of the line 
which cuts it in three points, whose distances are in arithmetical progression, is the 
curve 
A 3 + 2A, — 3A„ A, A 2 =0 ; ........ (9) 
and if U be a quartic, the envelope of the line which it cuts harmonically is the deter- 
minant 
Ao, A n A 2 
A„ A 2 , A 3 
A-2, A3, A 4 
=0, 
(10) 
7. Let the curve U=0 be a conic, then the equation (4) becomes 
(A 0 , a 15 a 2 )>, i) 2 =o. 
Now if A, = Q, it is evident the line y=(v — x)t will cut the curve in two points, which 
are equally distant from the axis of y ; but when n= 2, A! becomes 
v{a x [h— a.pC)— ^(b^— b<f)=0 ; (11) 
that is, a conic section. Hence we have the following theorems, the second of which is 
the projection of the first, and follows from the equation in X, p, u, as the first does from 
the corresponding one in v and t : — 
1st. If a variable line intersect a conic section , and if the locus of its middle point 
be a right line , its envelope is a conic section. 
2nd. If a variable line be cut harmonically by a conic section and a pair of lines , its 
envelope is a conic section touching the pair of lines. 
8. Let U be the cubic 
(a 0 , a,, a 2 , a 3 Xx, yf+ 3(5., b 2 , b 3 Jx, y) 2 +3(c 2 , c 3 Jx, y)+d s = 0, . . (12) 
and the curve A, will be 
»t(a 1, a„ asjl—ty+ibi, b 2 , b-Jf — 1) 2 =0 (13) 
This equation is the condition that the locus of the mean centre of the points where 
the line x-\-y cot <p~v meets the curve is the axis of y ; and since the axis of y may be 
any line, we have the following theorem : — If a variable line intersect a cubic in such 
a manner that the locus of the mean centre of the points where it meets the cubic is a 
right line , its envelope is a curve of the third class. 
9. The equation (13), expressed in the usual notation of tangential coordinates, is 
<«1, «2, ^ x) 2 — /*(&!, b 2 , b 3 Xp— *) 2 =0 (14) 
