[HARDING] RATIONAL PLANE ANHARMONIC CUBICS" 193 



In a similar manner we obtain 



(46) m'Km^8^ = 3Q-1200{mt-iyl{3yn + ôm)aa+{5^m-\-yl)ôb-{-2ôHc], 



(47) m'K^Hi^^ = 36- 1200 l*A{ama-^mb-\-ylb-8lc). 



Hence the coordinates of Pi, Po, Ps, the vertices of the invariant 

 triangle, are 



Xi^^ =lai-\-2mbi-\-nCi or lb^-{-2mc^-\-nd{, 



(48) xP = {'3'Yn+bm)aai+{b^m+yl)bb; + 2dHc;, (i=l, 2, 3) 

 3cP^ = ama, — (fim—yl)b[—ôlci, 



where the second value of .rp must be used if 5 = 0. 



A glance at (29) and (30) shows that Pi and Po are, respectively, 

 the cusp and the inflection point of the cubic. Writing x^^^ for x we 

 obtain 



\bax\ = —8l\bac\ =ôH, \cax\ = —{^m — yl)\cab\ = —(0m—yl)8, 



\bdx\ = ani\bda\ —ôl\bdc\ =Saym-\-aôl, 



\cdx\ =am\cda\ — (^m — yl)\cdb\ = Sa^m — (^m — yl) a = —d~n. 

 Equation (32) then reduces to 



{bU+anY{mt-l)=0. 

 That is, Ps is the point of intersection of the cuspidal tangent and the 

 inflectional tangent. Hence the invariant triangle of an anharmonic 

 cubic is the triangle formed by the cuspidal tangent, the inflectional 

 tangent, and the line joining the cusp to the point of inflection. 



10. Special Cases. Up to this point we have assumed that each 

 of the determinants /, m, n is diff'ercnt from zero. We are thus left with 

 two special cases to consider, namely: l = m = 0, W4=0, and m = n=0, 

 /^=0. It is easily shown that all our previous results hold in^each of 

 these special cases. We shall need the following identities: 



a» _ ,281 ^ , 25m 5/ ^^ , 2am 



— = 37+ — = 37+ , - = 3/3 + ——, 



/ m n n I 



(49) ^=-27-^^, li=_2/3-^, 

 m m m m 



L^ = -27-^^, ."-"=-2^-1-^. 



m l m m 



We shall state only the final results, which are easily obtained by 

 means of these identities. 



Case 1. l = m = 0, n^O. In this case — = - =0. Equations (5) 



m n 



show that a = /3 = 0, n = y'. Hence 



(4') 37Ci = ô^i. (i=l, 2, 3) 



(80 A^37/- + ô/^ 



