TRANSACTIONS OF THE SECTIONS. 27 



Let the equation be written with polar tangential coordinates 

 j>=aco8 (0+a) cos (0+/3) cos (6+y). 



The position of the cusps is thus determined : — Since -^=0, 



f =tan (0+/3) tan (6+y) + tan (6+y) tan (6+a)+ tan (6+a) tan (6+P). 



If the cusps form an equilateral triangle, the lines joining them with the satel- 

 lite-point pass through the foci. 



2. If two foci are at infinity, the line at infinity is a bitangent. 



The cubica of this division may be thus expressed by oblique Boothian co- 

 ordinates, 



^+0£+</7-l)(£ 2 +'? 2 -2£? cos »)=0. 



If X.ry+a; 2 -r-2/ 2 +2a^cos<i))=0, 



the cubic is resolved into a point and a parabola, 



\x y ) 



If X = 4 sin 2 ^-, or —4 cos 2 ", the cubic is inflexional, and acubitangential or veri- 



bitangential, according as X lies within or beyond these limits. 



The exhibited figures were constructed for rectangular coordinates. 



3. Let three single foci be collinear, and one of them at infinity. 

 Let the Boothian equation denote the group 



x$a-w+(*i+w-i)(£ 2 4y)=o, 



so that the origin bisects the distance between the foci. 



There are four critical lines, such that one of the coordinates is determined from 

 a quartic 



3&'£i _ 8& 2 *f +(26 2 + 4* 2 +4y 2 )£ 2 -1 = 0. 



Write r 2 for # 2 +y 2 , and 3s 2 for 2r 2 + b 2 ; the invariants are found to be 



S=3s 4 -36 4 ; T= -s G -36 4 s 2 +46V. 



(g\ 3 

 -j , deter- 

 mines the locus of the satellite-point (x, y), when the cubic is inflexional. 

 Its equation, when expanded, is an octavic, 



r 6(4r J -3r 2 )-6r 4 6 2 (r 2 -.r 2 )+36 4 (-3r 4 +10r 2 x 2 -9ar 4 )-i-26 6 (7.r 2 -3r 2 )-36 8 = 0. 



It denotes a unipartite curve whose asymptotes intersect at an angle 60°. For 

 such a position of the satellite on this locus there are three critical values of the 

 parameter, which yield inflexional, acubitangential, and veribitangential cubics. 

 If the satellite lies within the bounding curve, there are four critical values ; if be- 

 yond, two are real, and two imaginary. 



4. Let one focus be at infinity, and the quadrantal pole of the line joining the 

 other two. 



Let the equation to class-cubics of this system be written, 



X I? (l-^ 2 )+(^+M-l)(^+^)=0. 



There are five critical values, as may be proved by partial differentiation. The 

 following quintic determines the critical values of £ : — 



x - (4x 2 +4y 2 + J 2 )£ +* (4.r 2 +4y 2 +66 2 )f - 2b 2 (4a*. +b 2 )£ 3 +5& 4 .r| 4 - l 6 £ s = 0. 



The discriminant of this quintic determines the locus of the satellite, when the 

 cubic is inflexional. 



3* 



