594 
NOTES AND REFERENCES. 
solutions). It may be added that the 27 solutions form 9 groups of 3 each and that 
these 9 groups depend upon Hesse’s equation of the order 9 for the determination of 
the inflexions of a cubic curve ; and that the 12 solutions are determined by an 
equation of the order 12 which is the known resolvent of this order arising from 
Hesse’s equation and is solved by means of a quartic equation with a quadrinvariant 
= 0. As appears by the title of the memoir, the question is connected with that of 
the trisection of the hyperelliptic functions. 
401, 403. On the subject of Pascal’s theorem, see Veronese, “ Nuove teoremi sail’ 
hexagrammum mysticum,” R. Accad. dei Lincei (1870—77), pp. 7—61 ; Miss Christine 
Ladd (Mrs Franklin), “ The Pascal Hexagram,” Amer. Math. Jour., t. II. (1879), pp. 1—12, 
and Veronese, “ Interpretations géométriques de la théorie des substitutions de n lettres, 
particulièrement pour n = 3, 4, 5, en relation avec les groupes de l’Hexagramme Mysti 
que,” Ann. di Matem., t. xi. 1882—83, pp. 93—236. See also Richmond, “ A Sym 
metrical System of Equations of the Lines on a Cubic Surface which has a Conical 
Point,” Quart. Math. Jour., t. xxn. (1889), pp. 170—179, where the author discusses a 
perfectly symmetrical system of the lines on the cubic surface and deduces from them 
equations of the lines relating to a Pascal’s hexagon : there are of course through the 
conical point 6 lines lying on a quadric cone and these by their intersections with the 
plane give the six points of the hexagon : the interest of the paper consists as well 
in the connexion established between the two theories as in the perfectly symmetrical 
form given to the equations. 
406, 407. A correction was made by Halphen to the fundamental theorem of 
Chasles that the number of the conics (X, 4Z) is = a/x + /3i>, he finds that a diminution 
is in some cases required, and thus that the general form is, Number of conics 
(X, 4<Z) = ufjL + ¡3v — T : see Halphen’s two Notes, Comptes Rendus, 4 Sep. and 13 Nov., 
1876, t. lxxxiii. pp. 537 and 886, and his papers “Sur la théorie des caractéristiques 
pour les coniques,” Proc. Lond. Math. Soc., t. ix. (1877—1878), pp. 149—170, and “ Sur 
les nombres des coniques qui dans un plan satisfont à cinq conditions projectives et 
indépendantes entre elles,” Proc. Lond. Math. Soc., t. x. (1878—79), pp. 76—87 : also 
Zeuthen’s paper “ Sur la revision de la théorie des caractéristiques de M. Study,” 
Math. Ann., t. xxxvii. (1890), pp. 461—464, where the point is brought out very clearly 
and tersely. 
The correction rests upon a more complete development of the notion of the 
line-pair-point, viz. this degenerate form of conic seems at first sight to depend upon 
three parameters only, the two parameters which determine the position of the coincident 
lines, and a third parameter which determines the position therein of the coincident 
points : but there is really a fourth parameter. {Compare herewith the point-pair, or 
indefinitely thin conic, which working with point-coordinates presents itself in the first 
instance as a coincident line-pair depending on two parameters only, but which really 
depends also on the two parameters which determine the position therein of the vertices.] 
As to the fourth parameter of the line-pair-point the most simple definition is a 
metrical one; taking the semiaxes of the degenerate conic to be a and h (a = 0, 6 = 0) 
then we have two positive integers p and q prime to each other such that the ratio