J 
748] 
221 
748. 
ON THE BITANGENTS OF A QUARTIC. 
[From Salmons Higher Plane Curves, (3rd ed., 1879), pp. 387—389.] 
The equations of the 28 bitangents of a quartic curve were obtained in a very 
elegant form by Riemann in the paper “ Zur Theorie der Abel’schen Functionen fur 
den Fall p= 3,” Ges. Werke, Leipzig, 187G, pp. 456—472; and see also Weber’s Theorie 
der Abel’schen Functionen vom Geschlecht 3,” Berlin, 1876. Riemann connects the 
several bitangents with the characteristics of the 28 odd functions, thus obtaining for 
them an algorithm which it is worth while to explain, but they will be given also 
with the algorithm employed p. 231 et seq. of the present work*, which is in fact the 
more simple one. The characteristic of a triple 0-function is a symbol of the form 
a #7» 
where each of the letters is = 0 or 1; there are thus in all 64 such symbols, but they 
are considered as odd or even according as the sum aa + /3/3' + 77' is odd or even; 
and the numbers of the odd and even characteristics are 28 and 36 respectively; and, 
as already mentioned, the 28 odd characteristics correspond to the 28 bitangents 
respectively. 
We have x, y, z trilinear coordinates, a, /3, 7, ¡3<y' constants chosen at pleasure, 
and then a", /3", 7" determinate constants, such that the equations 
x+ y + z+ % + •»?+£’= 0, 
<zx + py + yz+£ + | + - = 0, 
a x + y + 7' z + + ^7 + 4 = 0, 
a"x + (3"y + 7" z + + = 
[* That is, Salmon’s Higher Plane Curves.]