454 
ON THE TRIPLE ^-FUNCTIONS. 
[702 
To each even characteristic there belongs a system of 56 hemi-tripairs; thus for 
the characteristic qqq , the 56 hemi-tripairs are 123, that is, 12, 13, 23, etc.: whence 
the 70 tripairs are 1234, that is, 12.34; 13.24; 14.23, etc.; and in any such 
tripair, say in 1234, we have the set of four hemi-tripairs 123, 124, 134, 234, for 
each of which the sum of the three characteristics is 
and the other set 1.234, 2.134, 3.124, 4.123, for each of which the sum of the 
three characteristics is 
To find the hemi-tripairs that belong to any other even characteristic; for instance, 
qqj , corresponds to 567 : we have 4 such as 1.234; 24 such as (5.12)34; 4 such as 
5.678; and 24 such as (1.56) 78 ; in all 4 + 24 + 4 + 24, = 56. The tripairs are the 
2, 1234, 5678; 16 such as 54(123); 16 such as 15(678); 36 such as (5162)34.78; 
in all 2 + 16+ 16+36, = 70; and in each of these it is easy to select the hemi- 
tripairs for which the sum of the 3 duads is = 567. 
Cambridge, 27 December, 1878.