444 ALGORITHM FOR THE CHARACTERISTICS OF THE TRIPLE ^-FUNCTIONS. [701 
by means of which the two-line-characteristic is at once found when the duad or 
triad is given. 
The new algorithm renders unnecessary the Table I. of Weber’s memoir “Theorie 
der Abel’schen Functionen vom Geschlecht 3” (Berlin, 1876). In fact, the system of 
six pairs corresponding to an odd characteristic such as 12 is 
13.23, 14.24, 15.25, 16.26, 17.27, 18.28, 
and that corresponding to an even characteristic such as 123 (= 1238.4567) is 
12.38, 13.28, 18.23, 45.67, 46.57, 47.56: 
so that all the (28 + 35 =) 63 systems can be at once formed. 
The odd characteristics correspond to the bitangents of a quartic curve, and as 
regards these bitangents the notation is, in fact, the notation arising out of Hesse’s 
investigations and explained Salmon’s Higher Plane Curves (2nd Ed. 1873), pp. 222—225. 
It may be noticed that the geometrical symbols corresponding to the before-mentioned 
two systems are: 
Hence, selecting out of the first system any two pairs, we have a symbol □ : but 
selecting out of the second system any two pairs, we have a symbol which is either 
□ or ||||; so that in each case (Salmon, p. 224) the four bitangents are such that 
the eight points of contact lie on a conic. 
The 28 bitangents of the general quartic curve 
V + V x.£ 2 + V x 3 ^ 3 = 0, 
represented by the equations given by Weber, l.c., pp. 100, 101, and taken in the order 
in which they are there written down, have for their duad-characteristics 
18, 28, 38, 23, 13, 12, 48, 14, 58, 15, 68, 16, 78, 17, 24, 34, 25, 35, 
26, 36, 27, 37, 67, 57, 56, 45, 46, 47 
respectively. Taking out of any one of the 63 systems three pairs of bitangents at 
pleasure, these give rise to an equation of the curve of a form such as 
V x£x + V + V X.£ z = 0,