264 report— 1879. 



already published, in which one or more foci are at an infinite distance, or two or 

 three foci unite to form a multiple focus. 1 



2. The locus of the satellite- point, when there are inflexional cubics in a family 

 of confocal groups, is material to the classification : it is obtained by eliminating 

 the parameter from the quartic and sextic invariants of the cubic equation to a 

 group. According to the position of the satellite-point on or within the several 

 convolutions of this locus, a confocal group may contain an odd or even number 

 of critical values ; if the satellite is on the locus, there is one inflexional cubic, 

 and there may be three or one other critical values ; and if it do not lie on the 

 locus, there is an even number, four, two, or none. If a focus be at infinity, there 

 is one additional critical cubic. If the satellite He on a side of ABO, there is a loss 

 of a critical value. There are at the most six critical values. 



3. The envelope of the stationary tangents of the inflexional cubics in a family 

 of groups of confocal cubics is a class-quartic. 



4. Let there be inflexional cubics in a family of groups of class-cubics, thus 

 denoted : 



2k abcpqr + (axp + byq + cz?-)'S(a 2 p 2 — 2bcqr cos A) = o ; 



the locus of the satellite (.v, y, z) is found, by equating the invariants to zero. For 

 brevity I, m n, denote cos A, cos B, cos C. 



S=- | YK-{la + mP + ny)Y - |~a 2 + 2 + y 2 + (2J + 4m?i)/3y + ... ."1 



— 12k 2 (7/3y + my a + na/3) 



+ 12k J a/3y(l +P + »i 2 + n 2 + \lmri) + a(/3 2 + y 2 ) (I + win) + ....1=0. 



T s - 8 j (k - 2la) n - - 2 [a 2 + (21 + 4mn)/3yl J ' 

 + 144 { (k - 21a) 2 - 2 [a 2 + (21 + 4»in)/3y~] I 

 x ■ - K 2 2/,3y + k [~a/3y( 1 + ttmn + Si 2 ) + 2a(/3 2 + y 8 ) {I + mn) 



- 864 K 3 a/3y + 432k 2 j - (2J/3y) a + 2 [Vy 2 + (2J + 4mn)a 2 /3yl | = o. 



These forms are equally true for spherical and plane geometry. But if the cubic 

 is plane, S and T may be simplified. 



S = - (k 2 - 2k2,Iu ~~X- 12K 2 2^/3y + 6K^- 2 2a/3y = o 

 T.-8(K 2 -2K2/a-|) 3 



-144(K 2 -2K2^a-|:)(K 2 2//3y-^2«/3y) 



-864K 3 a/3y + 108~(2fl/3y) 2 = 0. 



The eliminant of k is the locus of the satellite-point. This would not be useful 

 to calculate ; but the asymptotes, the intersections by the sides of ABO, and by 

 parallels to those sides, and intersections by the circle circumscribed about ABO 

 (2a/3y = o) can be obtained in serviceable forms, as well as the form of the curve at 

 the vertices of ABO. 



1 See Keports of British Association, and the Quarterly Journal of Mathematics 

 for 1876-8, in which last-named periodical the present memoir will be published in 

 extenso. 



