MISS C. A. SCOTT ON PLANE CUBICS. 
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from which 
i.e., 
Similarly 
and therefore 
From (iv) and (vi) 
{12x3] = {2Vlx}, 
{xl23] = {12Vx} . 
{212xx'3} = {lVl23x] 
{xl23] = {2xVx] . 
(12Vx} = {2x'l'x] . 
Now since xx are the foci of 11', 22', we have 
(iv). 
(v) , 
(vi) . 
(vii) . 
therefore (vii) becomes 
i.e., 
{12'x'x} = {l'2x'x}, 
{\'2xx] = {2x'l'x}, 
{2L'xx'} = [2x'l'x], 
i.e., is ecj^uianharmonic, and therefore {OK^/c] is equianharmonic; i.e., if 
three inflexional tangents be concurrent, the cubic is equianharmonic. 
Conversely, if the cubic be equianharmonic, the inflexional tangents are concurrent 
in threes. We know that {KK'^/c] is harmonic, and therefore 
== {IiHiIaL}.(viii), 
and for this special case {TK^'k] is equianharmonic, and therefore 
= {TiHgXx'j.(ix). 
Now by (viii) and similar relations, 
[KkKK'k'K] = {1231'2'3'],. (x). 
and t, T are the foci of the left hand side, x, x of the right. 
By means of (ix.), (v.), and (x.), 
[TKIck] = {12'x.x] 
= {xl23} 
== {rKkK}, 
where r is one of the pair t, r. Thus one of the two points t, t comes at T, and 
MDCCCXCIV.—A. 2 L 
