702] 
ON THE TRIPLE ^--FUNCTIONS. 
even characteristic (56 x 5 =) 280 tripairs, which are however 70 tripairs each taken 
4 times. A tripair contains in all (2 3 =) 8 hemi-tripairs, but these divide themselves 
into two sets each of 4 hemi-tripairs such that for each hemi-tripair of the first set 
the three characteristics have a given sum, and for each hemi-tripair of the second 
set the three characteristics have a different given sum. Hence considering the 70 
tripairs corresponding as above to a given even characteristic, in any one of the 70 
tripairs, there is a set of 4 hemi-tripairs such that in each of them the sum of the 
three characteristics is equal to the given even characteristic ; and taking the bitangents 
J» 9’ h t 0 correspond to any one of these hemi-tripairs, the bitangents which corre 
spond to the other three hemi-tripairs will be b, c, f; c, a, g and a, b, h respectively ; 
and we thus obtain from any one of these one and the same representation 
a. l '/bcf+ /3 Vcag + 7 Vabh + 8 \! fgh = 0 
of the cubic hexad. And the 70 tripairs give thus the 70 representations of the 
same cubic hexad. 
The whole number of hemi-tripairs is 36 x 56 = 2016: it may be remarked that 
there exists a system of 288 heptads, each of 7 odd characteristics such that selecting 
at pleasure any 3 characteristics out of the heptad, we obtain always a hemi-tripair : 
we have thus in all 288 x 35 = 10080 hemi-tripairs : this is = 2016 x 5, or we have 
the 2016 hemi-tripairs each taken 5 times. Weber’s Table II. exhibits 36 out of the 
288 heptads. 
I recall that in the algorithm derived from Hesse’s theory the bitangents are 
represented by the duads 12, 13, ..., 78 formed with the eight figures 1, 2, 3, 4, 5, 
6, 7, 8 ; these duads correspond to the odd characteristics as shown in the Table, 
and the table shows also triads corresponding to all the even characteristics except 
000 
000' 
Top line of characteristic. 
•a 
000 
100 
010 
110 
001 
101 
011 
111 
000 
236 
345 
137 
467 
156 
124 
257 
100 
237 
67 
136 
12 
157 
48 
256 
35 
010 
245 
127 
23 
68 
134 
357 
15 
47 
110 
126 
13 
78 
145 
356 
25 
46 
234 
001 
567 
146 
125 
247 
45 
17 
38 
26 
101 
147 
58 
246 
34 
16 
123 
27 
367 
Oil 
135 
347 
14 
57 
17 
36 
167 
456 
111 
346 
24 
56 
235 
37 
267 
457 
18