of Edinburgh, Session 1880 - 81 . 167 
manner as not to satisfy that law. For a complete investigation of 
the subject both symbols are required, 
§ 10. Three kinds of indices are required; one kind to express 
the number of times a relationship symbol is repeated ; a second 
to express the number of times one genus-relationship is present 
in a compound relationship ; and the third to express the Boolian 
index which denotes the combination of the same relationship 
with itself. The first of these may be called the order-index, and 
the second the degree-index. 
§11. The Order-index. — It follows from the preceding laws that 
c p+<i — c <i+p an q c -p-5 = c -(9+p) ? c p-« i s not — c~ q+p . 
The forms to which c p ' q can reduce are d p “ iy ~ {q ~ 1 \ c^ -2 ^' 2 *, 
c (p ~ sH«- 3) ; until one of the indices is reduced to 0. It is 
evident that a relationship of an odd order, if reducible, can 
reduce only to one of an odd order, and one of an even order only 
to one of an even order. 
It also follows from the preceding laws that ( c v ) q = (c ? ) p ; that 
(c n ) p - q = c np ~ nq ; but that (c p ~ q ) n is not = c pn ~ qn . For example (c 2 ) 2-1 
= c 2 c 2 e~ 2 = c 4-2 that is, grandchild of grandchild of grandparent 
= great great grandchild of grandparent. But (c 2_1 ) 2 = c 2-1 c 2-1 not 
= c 4-2 that is, grandchild of parent of grandchild of parent is not 
equivalent to great great grandchild of grandparent. 
§ 12. The Degree-index. — The nature of this index may be best 
explained by taking an example. The relationship of full brother 
or sister is compounded of the relationship of half brother or sister 
on the father’s side, and of half brother or sister on the mother’s 
side, and may be expressed by 
.2 
Here the introduction of the vinculum or some such sign is 
necessary to distinguish the index of the degree from the index of 
the order. I use the sign • to denote logical combination. The 
index of the degree is not the Boolian index (which satisfies the 
law x 2 — x), for the one quality is not wholly equivalent to the 
VOL. xi. 
Y 
