18 REPORT— 187G. 



Let the cubic be tbus denoted : — 



The critical Talue (k=5-145S) determines the bitangeutial cubic. The cubic is 

 bipartite or unipartite, according as /ci^<;5"1458 ; equiharmonicif k = 1"()7 or —"17; 

 harmonic if k= — -lo. 



There may be two real asymptotes, since there is one bitangent in this division. 



II. When the satellite-point is on the bounding sextic. 



In this case there are two critical values of the parameter, corresponding to the 

 bitangential and inflexional cubics, viz. 11-09 and -08. The curve is bipartite if 

 /c>ll-09. 



III. AMien the satellite is inside the quartic, and is (1) not collinear, and (2) is 

 collinear with the foci. 



(1) There are three bitangents when /c = 47, 'O?, 'OG. For higher values of k, 

 the curve is bipartite ; between 47 and '07, unipartite ; then bipartite and below 

 the bitangential curve unipartite. 



There may be four asjauptotes. 



(2) Let the cubic be thus denoted : — 



There are two bitangents with real and imaginary contact, as is thus shown : — 



4+A(27e=-18Jc-62)+4X^i^c=0. 

 For the inflexional genus, the discriminant gives the condition 



This may be resolved into two factors : — 



(h-c) (b-dcf. 



The first factor resolves the cubic into a point and a circle ; tlie second factor indir 

 cates the cissoid : 



(3+hiy+u\9+bi),j'=o. 



The satellite-point in this case is the ap.?e in the quartic bounding curve. 



IV. If the satellite-point be at infinity (1) not collinear, (2) collinear, with the 

 two foci. 



(1) There are two bitangential or a single inflexional, or no bitangential form, 

 according as the satellite lies within, upon, or beyond the quartic curve. One 

 asymptote connects the double focus with the satellite ; the other three concur in 

 the point (^ = |); the polar conic of the line at infinity degenerates into these 

 points. 



(2) There may be two asymptotes, which unite in a bitangent, for a special value 

 of the parameter. 



V. If the double focus is at infinity, (1) not collinear, (2) collinear with tlie 

 single focus and satellite-point. 



The cubic has in all cases a bitangent ; find for a particular value of the para- 

 meter two bitangents coincide in a stationary tangent at a point of inflexion. The 

 inflexional cubic in (2) is the semicubical parabola. 



YI. If the single focus is at infinity (I) not collinear, (2) collinear with the 

 double focus and satellite. 



There are two bitangential forms, but no inflexional case. The recijirocals in (2) 

 are Newton's defective hyperbola;, with diameters and double foci. 



\T.I. If the single focus and satellite-point are both at infinity. 



The curve is central and parabolic, with a cusp at infinity, but cannot have a 

 bitangent. Its single asymptote connects the double focus with the satellite. All 

 cubics are equiharmonic of the form 



It is thus denoted in Boothian coordinates : — 



The reciprocal is Newton's central species (?>8) . 



