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