20 REPORT — 1876. 



2. There is a sextic bounding curve, the locus of the satellite-point, when there 

 is a point of inflexion ; this curve is bipartite with an oval. If the satellite is 

 within the oval, four critical values of the parameter yield bitangential forms of the 

 cubic ; if the satellite is on the oval, one inflexional and two bitangential cubics ; 

 if the satellite is outside the oval, two bitangential cubics. Its equation, in Giider- 

 inaun's coordinates, is 



{(^ib^-+3)f-+ib+aXb+3x)Y=27(P+lXx-+f){(.v+by-+f-y-. 



If the two foci are a quadrant apart; 6 = oo , and the bounding sextic becomes 



3. All cubics with double foci have double cyclic arcs. 

 Let the liue-equatiou to these cubics be written 



3Ka^2}^q+Scr2(aY - 2bcrq cos A) = 0. 

 Its equivalent point^equation maj' be thus arranged : — 



(/32+ 2/3y cos A + y2) { _ 12j3y 'k^ ^_ 3^^2(8^32,/- _ a--f + 27«=|3= + 18a=/3y COS A 

 — 20a^y=COsC+36a/3-y cos B) +12k[— /S^ - n/3' COsB+a*COS A - a^y COS C 

 — 3ay3^y COS C + (- COS B-|-2 COS A cos C)a-'/S + (cos A— 2 COS B cos C) a'^/3' 



-(l+2cos-C)a=^y]}+I2(6Vr- (^^V {|3-yV+2/c(-^^y-4a/3^ cos B 



+4a^/3= cos A +3a2/3y -[-2a/3'V cos C) + (a' -l-2ai3 cos C -l-/?=)2 [ . 



But, if V denote the volume of the tetrahedron constituted by the centre of the 

 sphere and the angiilar points of the triangle of reference, 



(GV)= = 2(aV + 26c/3y cos a) 



= («a+t/3 cosc+cy cos6)--i-6V(/3-+2/3y cos A+y-). 



Hence if ;;=0 be a double focus, its quadrantal polar («a+&/3co3c— cy cos& = 0) is 

 a double cyclic arc. (See § 7.) 



The proposition seems to be susceptible of simple proof and of generalization. 



4. If a spherical curve have a multiple cyclic arc, it has at least a double focus. 

 Let the triangle of reference be trirectangular (which assumption does not aft'ect 



the generality of the proof), and let the quartic exhibit AB as a multiple cyclic 

 arc : — 



<^rz"+(a^"-±»/=+r^-)Xm=0. 



The terms may be thus grouped : — 



In this form the imaginary lines (a;+yi' = 0) are seen to meet AB in two coincident 

 points I, J ; the tangents at these points are these arcs CI, C J : their point of con- 

 ciu-rence is therefore a double focus. Tliis proof seems applicable only to tlie case 

 where the focus is the quadi-autal pole of the cyclic arc, the points I, J being in 

 this case the shadows of the circular points at infinity. 

 The argument may be also thus stated. The two lines 



(a; ±yi=0) 



are common tangents to the curve, and to the imaginary sphere {x^+y--\-z- = 0) at 

 their point of contact of a high order; their intersection is consequently a quadruple, 

 or, if real values alone are considered, a double focus, and might be a multiple focus 

 of a higher order. 



5. On equiharmonic or neutral cubics with double foci. The cu.sps are collinear, 

 and in this case S, an invariant of the cubic equation, =0. 



(4 cos= A— 3)k=+2(cos A+3cosB cos C + 2cos A cos^Ojfc + sin^ C = 0. 



There are two possible or coincident or impossible cases, according to the value of 

 the parameter. 



