FOEM OF TANGENTIAL EQUATION. 
373 
Observation. — The curve A 2 for a cubic has properties similar to A,. They differ 
only in that the lines CL and CN are interchanged, for CL is a single and CN a double 
tangent to A 2 . 
18. If U be the general curve of the nth. degree, A x = 0 gives the following theorem : — 
Given a curve of the nth degree , and two lines CL and CN, then if O, O' be two 
movable points on these lines , such that the polar line of O with respect to U may pass 
through O', the envelope of 00' will be a curve of the nth class , to which the line CL 
will be a multiple tangent of the order (n— 1). 
19. If in the equation for A, given in art. 4 we substitute for v its value yfxt, as in 
art. (15), we shall find the equation of 00' in the form 
at n +nbt n -'+ v ^^ + & c (19) 
Hence (see Salmon’s ‘Higher Curves,’ second edition, p. 66) we have 
v=n, gj=2(n—l), x=S(n— 2),j ^0) 
&=(w— 2)(w-3), r=i(n-l)(n-2), i= 0 . J 
All this will also follow from the propositions of the following articles, of which this 
and the preceding are special cases. 
20. We will now examine the general case A m =0. 
The equation A m =0 gives us the following theorem : — If U = 0 be a curve of the nth 
degree , and CL, CN two given lines, then if O, O' be two points taken on these lines , 
such that the mth polar of O with respect to XI passes through O', then the envelope of 
00' is the curve of the nth class A,„=0. 
21. The curve A m touches the line CL in ( n—m ) points and CN in m points. 
Demonstration. — Since the mth polar of O passes through O', the (n—m) th polar of 
O' passes through O. Hence we have two ways of generating the curve. Now let the 
point O' move along CN until it becomes consecutive to C, and it is evident that the 
(n — m) points in which its (n — m)th polar intersects CL will be points of contact of 
CL with A m . In like manner the m points in which the mth polar of a point conse- 
cutive to C on the line CL intersects the line CN will be points of contact. Hence the 
proposition is proved. 
m2 _ yi — Omn I 0)7)2 
Cor. The number of double tangents which A )n has = -- — . 
For the line CL is equivalent to 
(n—m) (n—m — 1) . 
2 double tangents, 
and the line CN to 
— m c) double tangents ; 
( 21 ) 
we have 
2r~n 2 — n — 2mn -f 2 m 2 
